Isotropic LOD (LOD)

Percentage Accurate: 68.2% → 68.2%

Time: 24.3s

Alternatives: 21

Speedup: 0.5×

Specification

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq w \land w \leq 16384\right) \land \left(1 \leq h \land h \leq 16384\right)\right) \land \left(1 \leq d \land d \leq 4096\right)\right) \land \left(10^{-20} \leq \left|dX.u\right| \land \left|dX.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.v\right| \land \left|dX.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dX.w\right| \land \left|dX.w\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.u\right| \land \left|dY.u\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.v\right| \land \left|dY.v\right| \leq 10^{+20}\right)\right) \land \left(10^{-20} \leq \left|dY.w\right| \land \left|dY.w\right| \leq 10^{+20}\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Initial Program: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
      (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = log2(sqrt(max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)))));
end

Alternative 1: 68.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := \mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)\\ \mathbf{if}\;t\_6 \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{t\_6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u))
        (t_6
         (fmax
          (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
          (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_3 t_3)))))
   (if (<= t_6 INFINITY)
     (log2 (sqrt t_6))
     (log2
      (sqrt
       (fmax
        (* (* dX.w dX.w) (pow (floor d) 2.0))
        (* dY.v (* dY.v (pow (floor h) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)));
	float tmp;
	if (t_6 <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(t_6));
	} else {
		tmp = log2f(sqrtf(fmaxf(((dX_46_w * dX_46_w) * powf(floorf(d), 2.0f)), (dY_46_v * (dY_46_v * powf(floorf(h), 2.0f))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = (Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))))
	tmp = Float32(0.0)
	if (t_6 <= Float32(Inf))
		tmp = log2(sqrt(t_6));
	else
		tmp = log2(sqrt(((Float32(Float32(dX_46_w * dX_46_w) * (floor(d) ^ Float32(2.0))) != Float32(Float32(dX_46_w * dX_46_w) * (floor(d) ^ Float32(2.0)))) ? Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0)))) : ((Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0)))) != Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0))))) ? Float32(Float32(dX_46_w * dX_46_w) * (floor(d) ^ Float32(2.0))) : max(Float32(Float32(dX_46_w * dX_46_w) * (floor(d) ^ Float32(2.0))), Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	t_6 = max((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + (t_1 * t_1)) + (t_3 * t_3)));
	tmp = single(0.0);
	if (t_6 <= single(Inf))
		tmp = log2(sqrt(t_6));
	else
		tmp = log2(sqrt(max(((dX_46_w * dX_46_w) * (floor(d) ^ single(2.0))), (dY_46_v * (dY_46_v * (floor(h) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))) < +inf.0

   1. Initial program 65.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   if +inf.0 < (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))

   1. Initial program 65.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(dY.v \cdot dY.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right)\right)}\right) \]

      7. lower-floor.f32 50.2

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

   6. Taylor expanded in dX.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot dX.w\right) \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      5. lower-floor.f32 32.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot dX.w\right) \cdot {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

   8. Applied rewrites 32.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_4 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_7 := \left\lfloor h\right\rfloor \cdot dY.v\\ \mathbf{if}\;dX.v \leq 200000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_6, dX.w \cdot \left(dX.w \cdot t\_3\right)\right), \left(t\_4 \cdot t\_4 + t\_7 \cdot t\_7\right) + t\_2 \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_0 \cdot t\_0\right) + t\_1 \cdot t\_1, \mathsf{fma}\left(dY.u, dY.u \cdot t\_6, dY.w \cdot \left(dY.w \cdot t\_3\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor d) dX.w))
        (t_2 (* (floor d) dY.w))
        (t_3 (pow (floor d) 2.0))
        (t_4 (* (floor w) dY.u))
        (t_5 (* (floor w) dX.u))
        (t_6 (pow (floor w) 2.0))
        (t_7 (* (floor h) dY.v)))
   (if (<= dX.v 200000.0)
     (log2
      (sqrt
       (fmax
        (fma dX.u (* dX.u t_6) (* dX.w (* dX.w t_3)))
        (+ (+ (* t_4 t_4) (* t_7 t_7)) (* t_2 t_2)))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_5 t_5) (* t_0 t_0)) (* t_1 t_1))
        (fma dY.u (* dY.u t_6) (* dY.w (* dY.w t_3)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(d) * dX_46_w;
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = powf(floorf(d), 2.0f);
	float t_4 = floorf(w) * dY_46_u;
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = powf(floorf(w), 2.0f);
	float t_7 = floorf(h) * dY_46_v;
	float tmp;
	if (dX_46_v <= 200000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_6), (dX_46_w * (dX_46_w * t_3))), (((t_4 * t_4) + (t_7 * t_7)) + (t_2 * t_2)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_0 * t_0)) + (t_1 * t_1)), fmaf(dY_46_u, (dY_46_u * t_6), (dY_46_w * (dY_46_w * t_3))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(d) * dX_46_w)
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = floor(d) ^ Float32(2.0)
	t_4 = Float32(floor(w) * dY_46_u)
	t_5 = Float32(floor(w) * dX_46_u)
	t_6 = floor(w) ^ Float32(2.0)
	t_7 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(200000.0))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_6), Float32(dX_46_w * Float32(dX_46_w * t_3))) != fma(dX_46_u, Float32(dX_46_u * t_6), Float32(dX_46_w * Float32(dX_46_w * t_3)))) ? Float32(Float32(Float32(t_4 * t_4) + Float32(t_7 * t_7)) + Float32(t_2 * t_2)) : ((Float32(Float32(Float32(t_4 * t_4) + Float32(t_7 * t_7)) + Float32(t_2 * t_2)) != Float32(Float32(Float32(t_4 * t_4) + Float32(t_7 * t_7)) + Float32(t_2 * t_2))) ? fma(dX_46_u, Float32(dX_46_u * t_6), Float32(dX_46_w * Float32(dX_46_w * t_3))) : max(fma(dX_46_u, Float32(dX_46_u * t_6), Float32(dX_46_w * Float32(dX_46_w * t_3))), Float32(Float32(Float32(t_4 * t_4) + Float32(t_7 * t_7)) + Float32(t_2 * t_2)))))));
	else
		tmp = log2(sqrt(((Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_1 * t_1))) ? fma(dY_46_u, Float32(dY_46_u * t_6), Float32(dY_46_w * Float32(dY_46_w * t_3))) : ((fma(dY_46_u, Float32(dY_46_u * t_6), Float32(dY_46_w * Float32(dY_46_w * t_3))) != fma(dY_46_u, Float32(dY_46_u * t_6), Float32(dY_46_w * Float32(dY_46_w * t_3)))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)), fma(dY_46_u, Float32(dY_46_u * t_6), Float32(dY_46_w * Float32(dY_46_w * t_3))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 2e5

   1. Initial program 69.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{dX.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      13. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \color{blue}{\left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \color{blue}{\left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      15. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      16. lower-floor.f32 63.2

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2}\right)\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 63.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   if 2e5 < dX.v

   1. Initial program 45.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 48.2

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 48.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 200000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_4 := \left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + t\_3 \cdot t\_3\\ \mathbf{if}\;dY.u \leq 0.17000000178813934:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_4, \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_4, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, t\_1\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* dY.w (* dY.w (pow (floor d) 2.0))))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dX.w))
        (t_4 (+ (+ (* t_0 t_0) (* t_2 t_2)) (* t_3 t_3))))
   (if (<= dY.u 0.17000000178813934)
     (log2 (sqrt (fmax t_4 (fma dY.v (* dY.v (pow (floor h) 2.0)) t_1))))
     (log2 (sqrt (fmax t_4 (fma dY.u (* dY.u (pow (floor w) 2.0)) t_1)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = dY_46_w * (dY_46_w * powf(floorf(d), 2.0f));
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dX_46_w;
	float t_4 = ((t_0 * t_0) + (t_2 * t_2)) + (t_3 * t_3);
	float tmp;
	if (dY_46_u <= 0.17000000178813934f) {
		tmp = log2f(sqrtf(fmaxf(t_4, fmaf(dY_46_v, (dY_46_v * powf(floorf(h), 2.0f)), t_1))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_4, fmaf(dY_46_u, (dY_46_u * powf(floorf(w), 2.0f)), t_1))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dX_46_w)
	t_4 = Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3))
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.17000000178813934))
		tmp = log2(sqrt(((t_4 != t_4) ? fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), t_1) : ((fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), t_1) != fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), t_1)) ? t_4 : max(t_4, fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), t_1))))));
	else
		tmp = log2(sqrt(((t_4 != t_4) ? fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), t_1) : ((fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), t_1) != fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), t_1)) ? t_4 : max(t_4, fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), t_1))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.170000002

   1. Initial program 64.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 58.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 58.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   if 0.170000002 < dY.u

   1. Initial program 68.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 68.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 68.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 0.17000000178813934:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_2 := \left\lfloor w\right\rfloor \cdot dX.u\\ \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_2 \cdot t\_2 + t\_0 \cdot t\_0\right) + t\_1 \cdot t\_1, \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor d) dX.w))
        (t_2 (* (floor w) dX.u)))
   (log2
    (sqrt
     (fmax
      (+ (+ (* t_2 t_2) (* t_0 t_0)) (* t_1 t_1))
      (fma
       dY.u
       (* dY.u (pow (floor w) 2.0))
       (* dY.w (* dY.w (pow (floor d) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(d) * dX_46_w;
	float t_2 = floorf(w) * dX_46_u;
	return log2f(sqrtf(fmaxf((((t_2 * t_2) + (t_0 * t_0)) + (t_1 * t_1)), fmaf(dY_46_u, (dY_46_u * powf(floorf(w), 2.0f)), (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f)))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(d) * dX_46_w)
	t_2 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((Float32(Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)) != Float32(Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0)) + Float32(t_1 * t_1))) ? fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) : ((fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) != fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))) ? Float32(Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)) : max(Float32(Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)), fma(dY_46_u, Float32(dY_46_u * (floor(w) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))))))))
end

Derivation

1. Initial program 65.6%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
2. Add Preprocessing

3. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

   8. lower-floor.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

   9. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

   10. associate-*l* N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

   11. *-commutative N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

   14. lower-pow.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

   15. lower-floor.f32 61.8

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

5. Applied rewrites 61.8%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

6. Final simplification 61.8%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right) \]
7. Add Preprocessing

Alternative 5: 56.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_3 := \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_2 \cdot t\_2\\ \mathbf{if}\;dY.w \leq 58000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_3, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_3, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor h) dX.v))
        (t_2 (* (floor d) dX.w))
        (t_3 (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_2 t_2))))
   (if (<= dY.w 58000000.0)
     (log2 (sqrt (fmax t_3 (* dY.u (* dY.u (pow (floor w) 2.0))))))
     (log2 (sqrt (fmax t_3 (* dY.w (* dY.w (pow (floor d) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = floorf(d) * dX_46_w;
	float t_3 = ((t_0 * t_0) + (t_1 * t_1)) + (t_2 * t_2);
	float tmp;
	if (dY_46_w <= 58000000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_3, (dY_46_u * (dY_46_u * powf(floorf(w), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_3, (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(d) * dX_46_w)
	t_3 = Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(58000000.0))
		tmp = log2(sqrt(((t_3 != t_3) ? Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))) : ((Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))) != Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) ? t_3 : max(t_3, Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt(((t_3 != t_3) ? Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) : ((Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) != Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) ? t_3 : max(t_3, Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(h) * dX_46_v;
	t_2 = floor(d) * dX_46_w;
	t_3 = ((t_0 * t_0) + (t_1 * t_1)) + (t_2 * t_2);
	tmp = single(0.0);
	if (dY_46_w <= single(58000000.0))
		tmp = log2(sqrt(max(t_3, (dY_46_u * (dY_46_u * (floor(w) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max(t_3, (dY_46_w * (dY_46_w * (floor(d) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 5.8e7

   1. Initial program 67.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(dY.u \cdot dY.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right)\right)}\right) \]

      7. lower-floor.f32 58.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right)\right)}\right) \]

   5. Applied rewrites 58.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

   if 5.8e7 < dY.w

   1. Initial program 53.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 56.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 56.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 58000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := dY.u \cdot t\_0\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dX.v \leq 50:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_0, \left(dX.w \cdot dX.w\right) \cdot t\_2\right), \mathsf{fma}\left(dY.u, t\_4, dY.w \cdot \left(dY.w \cdot t\_2\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, dY.u \cdot t\_4\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (* (floor w) dX.u))
        (t_2 (pow (floor d) 2.0))
        (t_3 (* (floor h) dX.v))
        (t_4 (* dY.u t_0))
        (t_5 (* (floor d) dX.w)))
   (if (<= dX.v 50.0)
     (log2
      (sqrt
       (fmax
        (fma dX.u (* dX.u t_0) (* (* dX.w dX.w) t_2))
        (fma dY.u t_4 (* dY.w (* dY.w t_2))))))
     (log2
      (sqrt
       (fmax (+ (+ (* t_1 t_1) (* t_3 t_3)) (* t_5 t_5)) (* dY.u t_4)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = powf(floorf(d), 2.0f);
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = dY_46_u * t_0;
	float t_5 = floorf(d) * dX_46_w;
	float tmp;
	if (dX_46_v <= 50.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_0), ((dX_46_w * dX_46_w) * t_2)), fmaf(dY_46_u, t_4, (dY_46_w * (dY_46_w * t_2))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_3 * t_3)) + (t_5 * t_5)), (dY_46_u * t_4))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = floor(d) ^ Float32(2.0)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(dY_46_u * t_0)
	t_5 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(50.0))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_2)) != fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_2))) ? fma(dY_46_u, t_4, Float32(dY_46_w * Float32(dY_46_w * t_2))) : ((fma(dY_46_u, t_4, Float32(dY_46_w * Float32(dY_46_w * t_2))) != fma(dY_46_u, t_4, Float32(dY_46_w * Float32(dY_46_w * t_2)))) ? fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_2)) : max(fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_2)), fma(dY_46_u, t_4, Float32(dY_46_w * Float32(dY_46_w * t_2))))))));
	else
		tmp = log2(sqrt(((Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)) != Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5))) ? Float32(dY_46_u * t_4) : ((Float32(dY_46_u * t_4) != Float32(dY_46_u * t_4)) ? Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)) : max(Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(dY_46_u * t_4))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 50

   1. Initial program 68.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 63.9

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 63.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 57.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   if 50 < dX.v

   1. Initial program 51.1%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(dY.u \cdot dY.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right)\right)}\right) \]

      7. lower-floor.f32 50.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right)\right)}\right) \]

   5. Applied rewrites 50.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 50:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_2 := \mathsf{fma}\left(dY.u, dY.u \cdot t\_0, dY.w \cdot \left(dY.w \cdot t\_1\right)\right)\\ \mathbf{if}\;dX.v \leq 5000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_0, \left(dX.w \cdot dX.w\right) \cdot t\_1\right), t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot t\_1\right)\right), t\_2\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (pow (floor d) 2.0))
        (t_2 (fma dY.u (* dY.u t_0) (* dY.w (* dY.w t_1)))))
   (if (<= dX.v 5000000.0)
     (log2 (sqrt (fmax (fma dX.u (* dX.u t_0) (* (* dX.w dX.w) t_1)) t_2)))
     (log2
      (sqrt
       (fmax
        (fma (* dX.v dX.v) (pow (floor h) 2.0) (* dX.w (* dX.w t_1)))
        t_2))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(d), 2.0f);
	float t_2 = fmaf(dY_46_u, (dY_46_u * t_0), (dY_46_w * (dY_46_w * t_1)));
	float tmp;
	if (dX_46_v <= 5000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_0), ((dX_46_w * dX_46_w) * t_1)), t_2)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_v * dX_46_v), powf(floorf(h), 2.0f), (dX_46_w * (dX_46_w * t_1))), t_2)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(d) ^ Float32(2.0)
	t_2 = fma(dY_46_u, Float32(dY_46_u * t_0), Float32(dY_46_w * Float32(dY_46_w * t_1)))
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(5000000.0))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)) != fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1))) ? t_2 : ((t_2 != t_2) ? fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)) : max(fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)), t_2)))));
	else
		tmp = log2(sqrt(((fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32(dX_46_w * Float32(dX_46_w * t_1))) != fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32(dX_46_w * Float32(dX_46_w * t_1)))) ? t_2 : ((t_2 != t_2) ? fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32(dX_46_w * Float32(dX_46_w * t_1))) : max(fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32(dX_46_w * Float32(dX_46_w * t_1))), t_2)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 5e6

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 64.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 64.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   if 5e6 < dX.v

   1. Initial program 41.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left({dX.v}^{2}, {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{dX.v \cdot dX.v}, {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{dX.v \cdot dX.v}, {\left(\left\lfloor h\right\rfloor \right)}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. *-inverses N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{1}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-rgt-identity N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      18. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, dX.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 43.9%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 5000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2}, dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_2 := \left(dX.w \cdot dX.w\right) \cdot t\_1\\ t_3 := \mathsf{fma}\left(dY.u, dY.u \cdot t\_0, dY.w \cdot \left(dY.w \cdot t\_1\right)\right)\\ \mathbf{if}\;dX.v \leq 5000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_0, t\_2\right), t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, t\_2\right), t\_3\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0))
        (t_1 (pow (floor d) 2.0))
        (t_2 (* (* dX.w dX.w) t_1))
        (t_3 (fma dY.u (* dY.u t_0) (* dY.w (* dY.w t_1)))))
   (if (<= dX.v 5000000.0)
     (log2 (sqrt (fmax (fma dX.u (* dX.u t_0) t_2) t_3)))
     (log2 (sqrt (fmax (fma dX.v (* dX.v (pow (floor h) 2.0)) t_2) t_3))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(d), 2.0f);
	float t_2 = (dX_46_w * dX_46_w) * t_1;
	float t_3 = fmaf(dY_46_u, (dY_46_u * t_0), (dY_46_w * (dY_46_w * t_1)));
	float tmp;
	if (dX_46_v <= 5000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_0), t_2), t_3)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_v, (dX_46_v * powf(floorf(h), 2.0f)), t_2), t_3)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(d) ^ Float32(2.0)
	t_2 = Float32(Float32(dX_46_w * dX_46_w) * t_1)
	t_3 = fma(dY_46_u, Float32(dY_46_u * t_0), Float32(dY_46_w * Float32(dY_46_w * t_1)))
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(5000000.0))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_0), t_2) != fma(dX_46_u, Float32(dX_46_u * t_0), t_2)) ? t_3 : ((t_3 != t_3) ? fma(dX_46_u, Float32(dX_46_u * t_0), t_2) : max(fma(dX_46_u, Float32(dX_46_u * t_0), t_2), t_3)))));
	else
		tmp = log2(sqrt(((fma(dX_46_v, Float32(dX_46_v * (floor(h) ^ Float32(2.0))), t_2) != fma(dX_46_v, Float32(dX_46_v * (floor(h) ^ Float32(2.0))), t_2)) ? t_3 : ((t_3 != t_3) ? fma(dX_46_v, Float32(dX_46_v * (floor(h) ^ Float32(2.0))), t_2) : max(fma(dX_46_v, Float32(dX_46_v * (floor(h) ^ Float32(2.0))), t_2), t_3)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 5e6

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 64.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 64.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   if 5e6 < dX.v

   1. Initial program 41.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, \color{blue}{dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, \color{blue}{dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. *-inverses N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{1}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 43.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 5000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_2 := dY.u \cdot t\_0\\ \mathbf{if}\;dX.v \leq 7000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_0, \left(dX.w \cdot dX.w\right) \cdot t\_1\right), \mathsf{fma}\left(dY.u, t\_2, dY.w \cdot \left(dY.w \cdot t\_1\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left(t\_1, dY.w \cdot dY.w, dY.u \cdot t\_2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (pow (floor d) 2.0)) (t_2 (* dY.u t_0)))
   (if (<= dX.v 7000000.0)
     (log2
      (sqrt
       (fmax
        (fma dX.u (* dX.u t_0) (* (* dX.w dX.w) t_1))
        (fma dY.u t_2 (* dY.w (* dY.w t_1))))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (fma t_1 (* dY.w dY.w) (* dY.u t_2))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(d), 2.0f);
	float t_2 = dY_46_u * t_0;
	float tmp;
	if (dX_46_v <= 7000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_0), ((dX_46_w * dX_46_w) * t_1)), fmaf(dY_46_u, t_2, (dY_46_w * (dY_46_w * t_1))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), fmaf(t_1, (dY_46_w * dY_46_w), (dY_46_u * t_2)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(d) ^ Float32(2.0)
	t_2 = Float32(dY_46_u * t_0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(7000000.0))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)) != fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1))) ? fma(dY_46_u, t_2, Float32(dY_46_w * Float32(dY_46_w * t_1))) : ((fma(dY_46_u, t_2, Float32(dY_46_w * Float32(dY_46_w * t_1))) != fma(dY_46_u, t_2, Float32(dY_46_w * Float32(dY_46_w * t_1)))) ? fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)) : max(fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)), fma(dY_46_u, t_2, Float32(dY_46_w * Float32(dY_46_w * t_1))))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * t_2)) : ((fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * t_2)) != fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * t_2))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * t_2)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 7e6

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 64.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 64.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 58.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   if 7e6 < dX.v

   1. Initial program 41.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 41.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 41.7%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   12. Applied rewrites 41.7%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 7000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.u, dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 5000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 5000000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor w) dX.u) 2.0)
      (+
       (pow (* (floor w) dY.u) 2.0)
       (+ (pow (* (floor h) dY.v) 2.0) (pow (* (floor d) dY.w) 2.0))))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (fma
       (pow (floor d) 2.0)
       (* dY.w dY.w)
       (* dY.u (* dY.u (pow (floor w) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 5000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (powf((floorf(w) * dY_46_u), 2.0f) + (powf((floorf(h) * dY_46_v), 2.0f) + powf((floorf(d) * dY_46_w), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), fmaf(powf(floorf(d), 2.0f), (dY_46_w * dY_46_w), (dY_46_u * (dY_46_u * powf(floorf(w), 2.0f)))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(5000000.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))) : ((Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))) != Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0))))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) + Float32((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? fma((floor(d) ^ Float32(2.0)), Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) : ((fma((floor(d) ^ Float32(2.0)), Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) != fma((floor(d) ^ Float32(2.0)), Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), fma((floor(d) ^ Float32(2.0)), Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 5e6

   1. Initial program 69.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 56.3

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 56.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. Applied rewrites 56.3%

      \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor w\right\rfloor \cdot dY.u\right)}^{2} + \left({\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)\right)}\right)} \]

   if 5e6 < dX.v

   1. Initial program 41.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 43.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 43.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 41.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 41.7%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   12. Applied rewrites 41.7%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 48.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := t\_0 \cdot \left(dY.w \cdot dY.w\right)\\ t_2 := \left\lfloor w\right\rfloor \cdot dY.u\\ \mathbf{if}\;dY.u \leq 0.014999999664723873:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot t\_0\right), t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \mathsf{fma}\left(t\_2, t\_2, t\_1\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0))
        (t_1 (* t_0 (* dY.w dY.w)))
        (t_2 (* (floor w) dY.u)))
   (if (<= dY.u 0.014999999664723873)
     (log2
      (sqrt
       (fmax
        (fma dX.u (* dX.u (pow (floor w) 2.0)) (* (* dX.w dX.w) t_0))
        t_1)))
     (log2
      (sqrt (fmax (* (pow (floor h) 2.0) (* dX.v dX.v)) (fma t_2 t_2 t_1)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float t_1 = t_0 * (dY_46_w * dY_46_w);
	float t_2 = floorf(w) * dY_46_u;
	float tmp;
	if (dY_46_u <= 0.014999999664723873f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * powf(floorf(w), 2.0f)), ((dX_46_w * dX_46_w) * t_0)), t_1)));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(h), 2.0f) * (dX_46_v * dX_46_v)), fmaf(t_2, t_2, t_1))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	t_1 = Float32(t_0 * Float32(dY_46_w * dY_46_w))
	t_2 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.014999999664723873))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(Float32(dX_46_w * dX_46_w) * t_0)) != fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(Float32(dX_46_w * dX_46_w) * t_0))) ? t_1 : ((t_1 != t_1) ? fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(Float32(dX_46_w * dX_46_w) * t_0)) : max(fma(dX_46_u, Float32(dX_46_u * (floor(w) ^ Float32(2.0))), Float32(Float32(dX_46_w * dX_46_w) * t_0)), t_1)))));
	else
		tmp = log2(sqrt(((Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)) != Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v))) ? fma(t_2, t_2, t_1) : ((fma(t_2, t_2, t_1) != fma(t_2, t_2, t_1)) ? Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)) : max(Float32((floor(h) ^ Float32(2.0)) * Float32(dX_46_v * dX_46_v)), fma(t_2, t_2, t_1))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.0149999997

   1. Initial program 65.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 59.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 59.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   9. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}}\right)}\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}}\right)}\right) \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dY.w}^{2}\right)}\right) \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot {dY.w}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)}\right)}\right) \]

       6. lower-*.f32 44.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)}\right)}\right) \]

   11. Applied rewrites 44.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)}\right)}\right) \]

   if 0.0149999997 < dY.u

   1. Initial program 66.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 67.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 67.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 51.1

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 53.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 53.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   11. Step-by-step derivation

       1. lift-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left({\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u\right) + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       2. lift-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u\right) + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       3. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       4. lift-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) + dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

       5. lift-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) + dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

       6. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) + dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

       7. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) + \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       9. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} \cdot dY.u + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       10. associate-*l* N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)} + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       11. lift-pow.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot \left(dY.u \cdot dY.u\right) + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       12. unpow2 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)} \cdot \left(dY.u \cdot dY.u\right) + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       13. unswap-sqr N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right)} + dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

       14. lower-fma.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor w\right\rfloor \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

       15. lower-*.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(\color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}, \left\lfloor w\right\rfloor \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

       16. lower-*.f32 53.1

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \color{blue}{\left\lfloor w\right\rfloor \cdot dY.u}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

       17. lift-*.f32 N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor w\right\rfloor \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

       18. *-commutative N/A

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor w\right\rfloor \cdot dY.u, \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right) \cdot dY.w}\right)\right)}\right) \]

   12. Applied rewrites 53.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor w\right\rfloor \cdot dY.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 0.014999999664723873:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \left(dX.v \cdot dX.v\right), \mathsf{fma}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor w\right\rfloor \cdot dY.u, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u \leq 0.014999999664723873:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot t\_0, \left(dX.w \cdot dX.w\right) \cdot t\_1\right), t\_1 \cdot \left(dY.w \cdot dY.w\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left(t\_1, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot t\_0\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (pow (floor d) 2.0)))
   (if (<= dY.u 0.014999999664723873)
     (log2
      (sqrt
       (fmax
        (fma dX.u (* dX.u t_0) (* (* dX.w dX.w) t_1))
        (* t_1 (* dY.w dY.w)))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (fma t_1 (* dY.w dY.w) (* dY.u (* dY.u t_0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(d), 2.0f);
	float tmp;
	if (dY_46_u <= 0.014999999664723873f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(dX_46_u, (dX_46_u * t_0), ((dX_46_w * dX_46_w) * t_1)), (t_1 * (dY_46_w * dY_46_w)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), fmaf(t_1, (dY_46_w * dY_46_w), (dY_46_u * (dY_46_u * t_0))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.014999999664723873))
		tmp = log2(sqrt(((fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)) != fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1))) ? Float32(t_1 * Float32(dY_46_w * dY_46_w)) : ((Float32(t_1 * Float32(dY_46_w * dY_46_w)) != Float32(t_1 * Float32(dY_46_w * dY_46_w))) ? fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)) : max(fma(dX_46_u, Float32(dX_46_u * t_0), Float32(Float32(dX_46_w * dX_46_w) * t_1)), Float32(t_1 * Float32(dY_46_w * dY_46_w)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_0))) : ((fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_0))) != fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), fma(t_1, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_0))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.0149999997

   1. Initial program 65.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 59.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 59.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2}} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dX.u}^{2} + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dX.u \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)} + {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \color{blue}{dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      13. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      14. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      17. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   9. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}}\right)}\right) \]

       2. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}}\right)}\right) \]

       3. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot {dY.w}^{2}\right)}\right) \]

       4. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot {dY.w}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)}\right)}\right) \]

       6. lower-*.f32 44.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.w \cdot dY.w\right)}\right)}\right) \]

   11. Applied rewrites 44.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)}\right)}\right) \]

   if 0.0149999997 < dY.u

   1. Initial program 66.6%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 67.0

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 67.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 51.1

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 51.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 53.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 53.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   12. Applied rewrites 53.1%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u \leq 0.17000000178813934:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_1\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot t\_0\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left(t\_0, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot t\_1\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)) (t_1 (pow (floor w) 2.0)))
   (if (<= dY.u 0.17000000178813934)
     (log2
      (sqrt
       (fmax
        (* dX.u (* dX.u t_1))
        (fma dY.v (* dY.v (pow (floor h) 2.0)) (* dY.w (* dY.w t_0))))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (fma t_0 (* dY.w dY.w) (* dY.u (* dY.u t_1)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float t_1 = powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_u <= 0.17000000178813934f) {
		tmp = log2f(sqrtf(fmaxf((dX_46_u * (dX_46_u * t_1)), fmaf(dY_46_v, (dY_46_v * powf(floorf(h), 2.0f)), (dY_46_w * (dY_46_w * t_0))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), fmaf(t_0, (dY_46_w * dY_46_w), (dY_46_u * (dY_46_u * t_1))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	t_1 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.17000000178813934))
		tmp = log2(sqrt(((Float32(dX_46_u * Float32(dX_46_u * t_1)) != Float32(dX_46_u * Float32(dX_46_u * t_1))) ? fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * t_0))) : ((fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * t_0))) != fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * t_0)))) ? Float32(dX_46_u * Float32(dX_46_u * t_1)) : max(Float32(dX_46_u * Float32(dX_46_u * t_1)), fma(dY_46_v, Float32(dY_46_v * (floor(h) ^ Float32(2.0))), Float32(dY_46_w * Float32(dY_46_w * t_0))))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_1))) : ((fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_1))) != fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_1)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * t_1))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.170000002

   1. Initial program 64.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 58.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 58.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. lower-floor.f32 42.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 42.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   if 0.170000002 < dY.u

   1. Initial program 68.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 68.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 68.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 52.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 52.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 53.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 53.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   12. Applied rewrites 53.8%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 0.17000000178813934:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \mathsf{fma}\left(dY.v, dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.u \leq 0.17000000178813934:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.v \cdot dY.v, dY.w \cdot \left(dY.w \cdot t\_0\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left(t\_0, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)))
   (if (<= dY.u 0.17000000178813934)
     (log2
      (sqrt
       (fmax
        (pow (* (floor w) dX.u) 2.0)
        (fma (pow (floor h) 2.0) (* dY.v dY.v) (* dY.w (* dY.w t_0))))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (fma t_0 (* dY.w dY.w) (* dY.u (* dY.u (pow (floor w) 2.0))))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float tmp;
	if (dY_46_u <= 0.17000000178813934f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), fmaf(powf(floorf(h), 2.0f), (dY_46_v * dY_46_v), (dY_46_w * (dY_46_w * t_0))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), fmaf(t_0, (dY_46_w * dY_46_w), (dY_46_u * (dY_46_u * powf(floorf(w), 2.0f)))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.17000000178813934))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? fma((floor(h) ^ Float32(2.0)), Float32(dY_46_v * dY_46_v), Float32(dY_46_w * Float32(dY_46_w * t_0))) : ((fma((floor(h) ^ Float32(2.0)), Float32(dY_46_v * dY_46_v), Float32(dY_46_w * Float32(dY_46_w * t_0))) != fma((floor(h) ^ Float32(2.0)), Float32(dY_46_v * dY_46_v), Float32(dY_46_w * Float32(dY_46_w * t_0)))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), fma((floor(h) ^ Float32(2.0)), Float32(dY_46_v * dY_46_v), Float32(dY_46_w * Float32(dY_46_w * t_0))))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) : ((fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) != fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.170000002

   1. Initial program 64.8%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot {dY.v}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.v \cdot dY.v\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 58.6

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 58.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]

   6. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. lower-floor.f32 42.7

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Applied rewrites 42.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \mathsf{fma}\left(dY.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 42.7%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor h\right\rfloor \right)}^{2}, dY.v \cdot dY.v, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]

   if 0.170000002 < dY.u

   1. Initial program 68.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 68.5

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 68.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 52.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 52.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 53.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 53.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   12. Applied rewrites 53.8%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u \leq 115000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left(t\_0, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot t\_0\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)))
   (if (<= dX.u 115000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (fma t_0 (* dY.w dY.w) (* dY.u (* dY.u (pow (floor w) 2.0)))))))
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 2.0) (* dY.w (* dY.w t_0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float tmp;
	if (dX_46_u <= 115000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), fmaf(t_0, (dY_46_w * dY_46_w), (dY_46_u * (dY_46_u * powf(floorf(w), 2.0f)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (dY_46_w * (dY_46_w * t_0)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(115000000.0))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) : ((fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))) != fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0)))))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), fma(t_0, Float32(dY_46_w * dY_46_w), Float32(dY_46_u * Float32(dY_46_u * (floor(w) ^ Float32(2.0))))))))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(dY_46_w * Float32(dY_46_w * t_0)) : ((Float32(dY_46_w * Float32(dY_46_w * t_0)) != Float32(dY_46_w * Float32(dY_46_w * t_0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(dY_46_w * Float32(dY_46_w * t_0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1.15e8

   1. Initial program 67.3%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right)} + {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u}, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      8. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot dY.u, {dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\right)}\right) \]

      11. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)\right)}\right) \]

      14. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)\right)}\right) \]

      15. lower-floor.f32 62.8

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   5. Applied rewrites 62.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)}\right) \]
   6. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)} \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\color{blue}{\left\lfloor h\right\rfloor } \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \color{blue}{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      6. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      7. lower-exp.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      8. rem-log-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{\log \left(e^{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot 2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      9. pow-to-exp N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\log \color{blue}{\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      10. log-pow N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{\color{blue}{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      12. lower-log.f32 48.2

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + e^{2 \cdot \color{blue}{\log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   7. Applied rewrites 48.2%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \color{blue}{e^{2 \cdot \log \left(\left\lfloor h\right\rfloor \cdot dX.v\right)}}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   8. Taylor expanded in dX.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      4. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

      5. lower-floor.f32 48.5

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.v \cdot dX.v\right) \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   10. Applied rewrites 48.5%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right) \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, \mathsf{fma}\left(dY.u, {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)\right)}\right) \]

   12. Applied rewrites 48.5%

       \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)} \]

   if 1.15e8 < dX.u

   1. Initial program 57.5%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 55.5

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 55.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 55.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

   8. Applied rewrites 55.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      6. unpow-prod-down N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      9. lower-pow.f32 55.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   10. Applied rewrites 55.0%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 115000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.w \cdot dY.w, dY.u \cdot \left(dY.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.w \leq 240:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.v \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.w 240.0)
   (log2
    (sqrt
     (fmax
      (+ (pow (* (floor h) dX.v) 2.0) (pow (* (floor d) dX.w) 2.0))
      (* dY.v (* dY.v (pow (floor h) 2.0))))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor w) dX.u) 2.0)
      (* dY.w (* dY.w (pow (floor d) 2.0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dY_46_w <= 240.0f) {
		tmp = log2f(sqrtf(fmaxf((powf((floorf(h) * dX_46_v), 2.0f) + powf((floorf(d) * dX_46_w), 2.0f)), (dY_46_v * (dY_46_v * powf(floorf(h), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(240.0))
		tmp = log2(sqrt(((Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) != Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w) ^ Float32(2.0)))) ? Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0)))) : ((Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0)))) != Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0))))) ? Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) : max(Float32((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) + (Float32(floor(d) * dX_46_w) ^ Float32(2.0))), Float32(dY_46_v * Float32(dY_46_v * (floor(h) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) : ((Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) != Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dY_46_w <= single(240.0))
		tmp = log2(sqrt(max((((floor(h) * dX_46_v) ^ single(2.0)) + ((floor(d) * dX_46_w) ^ single(2.0))), (dY_46_v * (dY_46_v * (floor(h) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (dY_46_w * (dY_46_w * (floor(d) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 240

   1. Initial program 67.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dY.v around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{{dY.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{\left(dY.v \cdot dY.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \left(\color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}} \cdot dY.v\right)\right)}\right) \]

      7. lower-floor.f32 53.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), dY.v \cdot \left({\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2} \cdot dY.v\right)\right)}\right) \]

   5. Applied rewrites 53.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \color{blue}{dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)}\right)}\right) \]

   6. Taylor expanded in dX.u around 0

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.v}^{2} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.v \cdot dX.v\right)} \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \color{blue}{\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right)} + {dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot 1}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      5. *-inverses N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{\frac{{dX.v}^{2}}{{dX.v}^{2}}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      6. associate-/l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) + \color{blue}{{dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      9. lower-fma.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.v, {\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right)}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      10. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, \color{blue}{dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, \color{blue}{dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      12. lower-pow.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      13. lower-floor.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}, {dX.v}^{2} \cdot \frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\frac{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}{{dX.v}^{2}} \cdot {dX.v}^{2}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\frac{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot {dX.v}^{2}}{{dX.v}^{2}}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      16. associate-/l* N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \color{blue}{\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \frac{{dX.v}^{2}}{{dX.v}^{2}}}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      17. *-inverses N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right) \cdot \color{blue}{1}\right), dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

   8. Applied rewrites 44.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\mathsf{fma}\left(dX.v, dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}, \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\color{blue}{\left(\left\lfloor h\right\rfloor \right)}}^{2}\right) + \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot \color{blue}{{\left(\left\lfloor h\right\rfloor \right)}^{2}}\right) + \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \color{blue}{\left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)} + \left(dX.w \cdot dX.w\right) \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + \color{blue}{{dX.w}^{2}} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      5. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + {dX.w}^{2} \cdot {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      6. rem-exp-log N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + {dX.w}^{2} \cdot {\color{blue}{\left(e^{\log \left(\left\lfloor d\right\rfloor \right)}\right)}}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      7. pow-prod-down N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + \color{blue}{{\left(dX.w \cdot e^{\log \left(\left\lfloor d\right\rfloor \right)}\right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      8. rem-exp-log N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + {\left(dX.w \cdot \color{blue}{\left\lfloor d\right\rfloor }\right)}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      9. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + {\color{blue}{\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + {\color{blue}{\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}}^{2}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right)}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      12. lift-*.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + \color{blue}{\left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right)}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.v \cdot \left(dX.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right)}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

   10. Applied rewrites 44.4%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}}, dY.v \cdot \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right)\right)}\right) \]

   if 240 < dY.w

   1. Initial program 58.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 56.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 56.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 51.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

   8. Applied rewrites 51.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      6. unpow-prod-down N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      9. lower-pow.f32 51.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   10. Applied rewrites 51.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 240:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2} + {\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, dY.v \cdot \left(dY.v \cdot {\left(\left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 240:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot t\_0\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot t\_0\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor d) 2.0)))
   (if (<= dY.w 240.0)
     (log2
      (sqrt
       (fmax (* dX.w (* dX.w t_0)) (* (pow (floor w) 2.0) (* dY.u dY.u)))))
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 2.0) (* dY.w (* dY.w t_0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(d), 2.0f);
	float tmp;
	if (dY_46_w <= 240.0f) {
		tmp = log2f(sqrtf(fmaxf((dX_46_w * (dX_46_w * t_0)), (powf(floorf(w), 2.0f) * (dY_46_u * dY_46_u)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (dY_46_w * (dY_46_w * t_0)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(240.0))
		tmp = log2(sqrt(((Float32(dX_46_w * Float32(dX_46_w * t_0)) != Float32(dX_46_w * Float32(dX_46_w * t_0))) ? Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u)) : ((Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u)) != Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u))) ? Float32(dX_46_w * Float32(dX_46_w * t_0)) : max(Float32(dX_46_w * Float32(dX_46_w * t_0)), Float32((floor(w) ^ Float32(2.0)) * Float32(dY_46_u * dY_46_u)))))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(dY_46_w * Float32(dY_46_w * t_0)) : ((Float32(dY_46_w * Float32(dY_46_w * t_0)) != Float32(dY_46_w * Float32(dY_46_w * t_0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(dY_46_w * Float32(dY_46_w * t_0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(240.0))
		tmp = log2(sqrt(max((dX_46_w * (dX_46_w * t_0)), ((floor(w) ^ single(2.0)) * (dY_46_u * dY_46_u)))));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (dY_46_w * (dY_46_w * t_0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 240

   1. Initial program 67.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \color{blue}{\left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \color{blue}{\left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 51.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 51.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      4. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot {dY.u}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)}\right) \]

      6. lower-*.f32 39.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)}\right) \]

   8. Applied rewrites 39.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)}\right)}\right) \]

   if 240 < dY.w

   1. Initial program 58.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 56.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 56.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 51.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

   8. Applied rewrites 51.8%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      6. unpow-prod-down N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      9. lower-pow.f32 51.8

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   10. Applied rewrites 51.8%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 240:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ t_1 := {\left(\left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.w \leq 1000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_0\right), t\_0 \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot t\_1\right), dY.w \cdot \left(dY.w \cdot t\_1\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)) (t_1 (pow (floor d) 2.0)))
   (if (<= dX.w 1000.0)
     (log2 (sqrt (fmax (* dX.u (* dX.u t_0)) (* t_0 (* dY.u dY.u)))))
     (log2 (sqrt (fmax (* dX.w (* dX.w t_1)) (* dY.w (* dY.w t_1))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float t_1 = powf(floorf(d), 2.0f);
	float tmp;
	if (dX_46_w <= 1000.0f) {
		tmp = log2f(sqrtf(fmaxf((dX_46_u * (dX_46_u * t_0)), (t_0 * (dY_46_u * dY_46_u)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((dX_46_w * (dX_46_w * t_1)), (dY_46_w * (dY_46_w * t_1)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	t_1 = floor(d) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(1000.0))
		tmp = log2(sqrt(((Float32(dX_46_u * Float32(dX_46_u * t_0)) != Float32(dX_46_u * Float32(dX_46_u * t_0))) ? Float32(t_0 * Float32(dY_46_u * dY_46_u)) : ((Float32(t_0 * Float32(dY_46_u * dY_46_u)) != Float32(t_0 * Float32(dY_46_u * dY_46_u))) ? Float32(dX_46_u * Float32(dX_46_u * t_0)) : max(Float32(dX_46_u * Float32(dX_46_u * t_0)), Float32(t_0 * Float32(dY_46_u * dY_46_u)))))));
	else
		tmp = log2(sqrt(((Float32(dX_46_w * Float32(dX_46_w * t_1)) != Float32(dX_46_w * Float32(dX_46_w * t_1))) ? Float32(dY_46_w * Float32(dY_46_w * t_1)) : ((Float32(dY_46_w * Float32(dY_46_w * t_1)) != Float32(dY_46_w * Float32(dY_46_w * t_1))) ? Float32(dX_46_w * Float32(dX_46_w * t_1)) : max(Float32(dX_46_w * Float32(dX_46_w * t_1)), Float32(dY_46_w * Float32(dY_46_w * t_1)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ single(2.0);
	t_1 = floor(d) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_w <= single(1000.0))
		tmp = log2(sqrt(max((dX_46_u * (dX_46_u * t_0)), (t_0 * (dY_46_u * dY_46_u)))));
	else
		tmp = log2(sqrt(max((dX_46_w * (dX_46_w * t_1)), (dY_46_w * (dY_46_w * t_1)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 1e3

   1. Initial program 67.0%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 55.9

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 55.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      4. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot {dY.u}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)}\right) \]

      6. lower-*.f32 39.4

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)}\right) \]

   8. Applied rewrites 39.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)}\right)}\right) \]

   if 1e3 < dX.w

   1. Initial program 60.4%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.w \cdot dX.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \color{blue}{\left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \color{blue}{\left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot \color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 55.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 55.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 51.0

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

   8. Applied rewrites 51.0%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 1000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.w \cdot \left(dX.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right), dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 38.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 58000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot t\_0\right), t\_0 \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor w) 2.0)))
   (if (<= dY.w 58000000.0)
     (log2 (sqrt (fmax (* dX.u (* dX.u t_0)) (* t_0 (* dY.u dY.u)))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor w) dX.u) 2.0)
        (* dY.w (* dY.w (pow (floor d) 2.0)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_w <= 58000000.0f) {
		tmp = log2f(sqrtf(fmaxf((dX_46_u * (dX_46_u * t_0)), (t_0 * (dY_46_u * dY_46_u)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(58000000.0))
		tmp = log2(sqrt(((Float32(dX_46_u * Float32(dX_46_u * t_0)) != Float32(dX_46_u * Float32(dX_46_u * t_0))) ? Float32(t_0 * Float32(dY_46_u * dY_46_u)) : ((Float32(t_0 * Float32(dY_46_u * dY_46_u)) != Float32(t_0 * Float32(dY_46_u * dY_46_u))) ? Float32(dX_46_u * Float32(dX_46_u * t_0)) : max(Float32(dX_46_u * Float32(dX_46_u * t_0)), Float32(t_0 * Float32(dY_46_u * dY_46_u)))))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) : ((Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) != Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(58000000.0))
		tmp = log2(sqrt(max((dX_46_u * (dX_46_u * t_0)), (t_0 * (dY_46_u * dY_46_u)))));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (dY_46_w * (dY_46_w * (floor(d) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 5.8e7

   1. Initial program 67.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 52.5

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 52.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot {dY.u}^{2}}\right)}\right) \]

      3. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}} \cdot {dY.u}^{2}\right)}\right) \]

      4. lower-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2} \cdot {dY.u}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)}\right) \]

      6. lower-*.f32 38.9

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \color{blue}{\left(dY.u \cdot dY.u\right)}\right)}\right) \]

   8. Applied rewrites 38.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)}\right)}\right) \]

   if 5.8e7 < dY.w

   1. Initial program 53.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in dX.u around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      7. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

      8. lower-floor.f32 49.9

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Applied rewrites 49.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. Taylor expanded in dY.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

      7. lower-floor.f32 53.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

   8. Applied rewrites 53.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-floor.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      5. lift-pow.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      6. unpow-prod-down N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

      9. lower-pow.f32 53.1

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   10. Applied rewrites 53.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 58000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), {\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot \left(dY.u \cdot dY.u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax (pow (* (floor w) dX.u) 2.0) (* (pow (floor d) 2.0) (* dY.w dY.w))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (powf(floorf(d), 2.0f) * (dY_46_w * dY_46_w)))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32((floor(d) ^ Float32(2.0)) * Float32(dY_46_w * dY_46_w)) : ((Float32((floor(d) ^ Float32(2.0)) * Float32(dY_46_w * dY_46_w)) != Float32((floor(d) ^ Float32(2.0)) * Float32(dY_46_w * dY_46_w))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32((floor(d) ^ Float32(2.0)) * Float32(dY_46_w * dY_46_w)))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), ((floor(d) ^ single(2.0)) * (dY_46_w * dY_46_w)))));
end

Derivation

1. Initial program 65.6%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
2. Add Preprocessing

3. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   8. lower-floor.f32 52.1

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

5. Applied rewrites 52.1%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

6. Taylor expanded in dY.w around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

   7. lower-floor.f32 35.8

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

8. Applied rewrites 35.8%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
9. Step-by-step derivation

   1. lift-floor.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   2. lift-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   4. pow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   6. unpow-prod-down N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   9. lower-pow.f32 35.8

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

10. Applied rewrites 35.8%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]
11. Step-by-step derivation

    1. lift-floor.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\color{blue}{\left\lfloor w\right\rfloor } \cdot dX.u\right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    2. lift-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    3. pow-to-exp N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{e^{\log \left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot 2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    4. lift-log.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{\color{blue}{\log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)} \cdot 2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    5. *-commutative N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{\color{blue}{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    6. lift-*.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{\color{blue}{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    7. lift-exp.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

    8. lift-floor.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}, dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

    9. lift-pow.f32 N/A

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}, dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

    10. lift-*.f32 N/A

        \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}, dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

    11. lift-*.f32 N/A

        \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}, \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

    12. lift-fmax.f32 N/A

        \[\leadsto \log_{2} \left(\sqrt{\color{blue}{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}}\right) \]

    13. lift-sqrt.f32 N/A

        \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloor w\right\rfloor \cdot dX.u\right)}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right)} \]

12. Applied rewrites 35.8%

    \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot \left(dY.w \cdot dY.w\right)\right)}\right)} \]
13. Add Preprocessing

Alternative 21: 35.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax (pow (* (floor w) dX.u) 2.0) (* dY.w (* dY.w (pow (floor d) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), (dY_46_w * (dY_46_w * powf(floorf(d), 2.0f))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) : ((Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))) != Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0))))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), Float32(dY_46_w * Float32(dY_46_w * (floor(d) ^ Float32(2.0)))))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (dY_46_w * (dY_46_w * (floor(d) ^ single(2.0)))))));
end

Derivation

1. Initial program 65.6%

   \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
2. Add Preprocessing

3. Taylor expanded in dX.u around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right)} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \color{blue}{\left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

   8. lower-floor.f32 52.1

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

5. Applied rewrites 52.1%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right)}, \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]

6. Taylor expanded in dY.w around inf

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{{dY.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}}\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{\left(dY.w \cdot dY.w\right)} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)}\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \color{blue}{\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left(\color{blue}{{\left(\left\lfloor d\right\rfloor \right)}^{2}} \cdot dY.w\right)\right)}\right) \]

   7. lower-floor.f32 35.8

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), dY.w \cdot \left({\color{blue}{\left(\left\lfloor d\right\rfloor \right)}}^{2} \cdot dY.w\right)\right)}\right) \]

8. Applied rewrites 35.8%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}\right), \color{blue}{dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)}\right)}\right) \]
9. Step-by-step derivation

   1. lift-floor.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot {\color{blue}{\left(\left\lfloor w\right\rfloor \right)}}^{2}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   2. lift-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(dX.u \cdot \left(dX.u \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}\right), dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{\left(dX.u \cdot dX.u\right) \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   4. pow2 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{dX.u}^{2}} \cdot {\left(\left\lfloor w\right\rfloor \right)}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   5. lift-pow.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot \color{blue}{{\left(\left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   6. unpow-prod-down N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   7. *-commutative N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}}^{2}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

   9. lower-pow.f32 35.8

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

10. Applied rewrites 35.8%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}}, dY.w \cdot \left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w\right)\right)}\right) \]

11. Final simplification 35.8%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, dY.w \cdot \left(dY.w \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}\right)\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :name "Isotropic LOD (LOD)"
  :precision binary32
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 w) (<= w 16384.0)) (and (<= 1.0 h) (<= h 16384.0))) (and (<= 1.0 d) (<= d 4096.0))) (and (<= 1e-20 (fabs dX.u)) (<= (fabs dX.u) 1e+20))) (and (<= 1e-20 (fabs dX.v)) (<= (fabs dX.v) 1e+20))) (and (<= 1e-20 (fabs dX.w)) (<= (fabs dX.w) 1e+20))) (and (<= 1e-20 (fabs dY.u)) (<= (fabs dY.u) 1e+20))) (and (<= 1e-20 (fabs dY.v)) (<= (fabs dY.v) 1e+20))) (and (<= 1e-20 (fabs dY.w)) (<= (fabs dY.w) 1e+20)))
  (log2 (sqrt (fmax (+ (+ (* (* (floor w) dX.u) (* (floor w) dX.u)) (* (* (floor h) dX.v) (* (floor h) dX.v))) (* (* (floor d) dX.w) (* (floor d) dX.w))) (+ (+ (* (* (floor w) dY.u) (* (floor w) dY.u)) (* (* (floor h) dY.v) (* (floor h) dY.v))) (* (* (floor d) dY.w) (* (floor d) dY.w)))))))