Isotropic LOD (LOD)

Percentage Accurate: 67.7% → 67.7%

Time: 20.8s

Alternatives: 10

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: 67.7% 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: 67.7% accurate, 0.5× speedup

Math

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

FPCore

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

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 = dY_46_v * floorf(h);
	float t_1 = dX_46_v * floorf(h);
	float t_2 = dY_46_u * floorf(w);
	float t_3 = dX_46_u * floorf(w);
	float t_4 = dY_46_w * floorf(d);
	float t_5 = dX_46_w * floorf(d);
	float t_6 = t_4 * t_4;
	float t_7 = fmaxf(((t_5 * t_5) + ((t_1 * t_1) + (t_3 * t_3))), (t_6 + ((t_0 * t_0) + (t_2 * t_2))));
	float tmp;
	if (t_7 <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(t_7));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_v * dX_46_v), powf(floorf(h), 2.0f), (powf(t_5, 2.0f) + powf(t_3, 2.0f))), (fmaf(t_2, t_2, powf(t_0, 2.0f)) + t_6))));
	}
	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(dY_46_v * floor(h))
	t_1 = Float32(dX_46_v * floor(h))
	t_2 = Float32(dY_46_u * floor(w))
	t_3 = Float32(dX_46_u * floor(w))
	t_4 = Float32(dY_46_w * floor(d))
	t_5 = Float32(dX_46_w * floor(d))
	t_6 = Float32(t_4 * t_4)
	t_7 = (Float32(Float32(t_5 * t_5) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3))) != Float32(Float32(t_5 * t_5) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)))) ? Float32(t_6 + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2))) : ((Float32(t_6 + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2))) != Float32(t_6 + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)))) ? Float32(Float32(t_5 * t_5) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3))) : max(Float32(Float32(t_5 * t_5) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3))), Float32(t_6 + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)))))
	tmp = Float32(0.0)
	if (t_7 <= Float32(Inf))
		tmp = log2(sqrt(t_7));
	else
		tmp = log2(sqrt(((fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32((t_5 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))) != fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32((t_5 ^ Float32(2.0)) + (t_3 ^ Float32(2.0))))) ? Float32(fma(t_2, t_2, (t_0 ^ Float32(2.0))) + t_6) : ((Float32(fma(t_2, t_2, (t_0 ^ Float32(2.0))) + t_6) != Float32(fma(t_2, t_2, (t_0 ^ Float32(2.0))) + t_6)) ? fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32((t_5 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))) : max(fma(Float32(dX_46_v * dX_46_v), (floor(h) ^ Float32(2.0)), Float32((t_5 ^ Float32(2.0)) + (t_3 ^ Float32(2.0)))), Float32(fma(t_2, t_2, (t_0 ^ Float32(2.0))) + t_6))))));
	end
	return 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 63.8%

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

   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 63.8%

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

      1. lift-+.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. unpow-prod-down N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

   4. Applied rewrites 44.3%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. pow-to-exp N/A

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

      7. lift-log.f32 N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 27.3

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

   6. Applied rewrites 27.3%

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-pow.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. *-commutative N/A

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

      7. lift-log.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-floor.f32 N/A

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

      11. pow-to-exp N/A

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

      12. pow2 N/A

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

      13. lower-fma.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f32 N/A

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

      20. pow2 N/A

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

   8. Applied rewrites 27.7%

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

4. Final simplification 63.8%

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

Alternative 2: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_3 := dY.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dY.w \leq 0.0020000000949949026:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), \left(t\_2 \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_2 \cdot dX.v\right) \cdot dX.v, t\_3 \cdot t\_3 + \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h)))
        (t_1 (* dY.u (floor w)))
        (t_2 (pow (floor h) 2.0))
        (t_3 (* dY.w (floor d))))
   (if (<= dY.w 0.0020000000949949026)
     (log2
      (sqrt
       (fmax
        (+
         (pow (* dX.v (floor h)) 2.0)
         (+ (pow (* dX.w (floor d)) 2.0) (pow (* dX.u (floor w)) 2.0)))
        (* (* t_2 dY.v) dY.v))))
     (log2
      (sqrt
       (fmax
        (* (* t_2 dX.v) dX.v)
        (+ (* t_3 t_3) (+ (* t_0 t_0) (* t_1 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 = dY_46_v * floorf(h);
	float t_1 = dY_46_u * floorf(w);
	float t_2 = powf(floorf(h), 2.0f);
	float t_3 = dY_46_w * floorf(d);
	float tmp;
	if (dY_46_w <= 0.0020000000949949026f) {
		tmp = log2f(sqrtf(fmaxf((powf((dX_46_v * floorf(h)), 2.0f) + (powf((dX_46_w * floorf(d)), 2.0f) + powf((dX_46_u * floorf(w)), 2.0f))), ((t_2 * dY_46_v) * dY_46_v))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((t_2 * dX_46_v) * dX_46_v), ((t_3 * t_3) + ((t_0 * t_0) + (t_1 * 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(dY_46_v * floor(h))
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = floor(h) ^ Float32(2.0)
	t_3 = Float32(dY_46_w * floor(d))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.0020000000949949026))
		tmp = log2(sqrt(((Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) != Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))))) ? Float32(Float32(t_2 * dY_46_v) * dY_46_v) : ((Float32(Float32(t_2 * dY_46_v) * dY_46_v) != Float32(Float32(t_2 * dY_46_v) * dY_46_v)) ? Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) : max(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))), Float32(Float32(t_2 * dY_46_v) * dY_46_v))))));
	else
		tmp = log2(sqrt(((Float32(Float32(t_2 * dX_46_v) * dX_46_v) != Float32(Float32(t_2 * dX_46_v) * dX_46_v)) ? Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))) : ((Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))) != Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)))) ? Float32(Float32(t_2 * dX_46_v) * dX_46_v) : max(Float32(Float32(t_2 * dX_46_v) * dX_46_v), Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * 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 = dY_46_v * floor(h);
	t_1 = dY_46_u * floor(w);
	t_2 = floor(h) ^ single(2.0);
	t_3 = dY_46_w * floor(d);
	tmp = single(0.0);
	if (dY_46_w <= single(0.0020000000949949026))
		tmp = log2(sqrt(max((((dX_46_v * floor(h)) ^ single(2.0)) + (((dX_46_w * floor(d)) ^ single(2.0)) + ((dX_46_u * floor(w)) ^ single(2.0)))), ((t_2 * dY_46_v) * dY_46_v))));
	else
		tmp = log2(sqrt(max(((t_2 * dX_46_v) * dX_46_v), ((t_3 * t_3) + ((t_0 * t_0) + (t_1 * t_1))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.00200000009

   1. Initial program 65.2%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 52.5

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 54.2

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

   10. Applied rewrites 54.2%

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

   if 0.00200000009 < dY.w

   1. Initial program 60.6%

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

   3. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 50.9

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

   5. Applied rewrites 50.9%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 0.0020000000949949026:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left(t\_0 + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0)))
   (if (<= dX.v 5.000000058430487e-8)
     (log2
      (sqrt
       (fmax
        t_0
        (+
         (pow (* dY.w (floor d)) 2.0)
         (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 2.0))))))
     (log2
      (sqrt
       (fmax
        (+ (pow (* dX.v (floor h)) 2.0) (+ t_0 (pow (* dX.u (floor w)) 2.0)))
        (* (* (pow (floor h) 2.0) dY.v) dY.v)))))))

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((dX_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_v <= 5.000000058430487e-8f) {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf((dY_46_w * floorf(d)), 2.0f) + (powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf((dX_46_v * floorf(h)), 2.0f) + (t_0 + powf((dX_46_u * floorf(w)), 2.0f))), ((powf(floorf(h), 2.0f) * dY_46_v) * dY_46_v))));
	}
	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(dX_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(5.000000058430487e-8))
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) : ((Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) != Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))))) ? t_0 : max(t_0, Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt(((Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_0 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) != Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_0 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))))) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v) : ((Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v)) ? Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_0 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) : max(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_0 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))), Float32(Float32((floor(h) ^ Float32(2.0)) * dY_46_v) * dY_46_v))))));
	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 = (dX_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(5.000000058430487e-8))
		tmp = log2(sqrt(max(t_0, (((dY_46_w * floor(d)) ^ single(2.0)) + (((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max((((dX_46_v * floor(h)) ^ single(2.0)) + (t_0 + ((dX_46_u * floor(w)) ^ single(2.0)))), (((floor(h) ^ single(2.0)) * dY_46_v) * dY_46_v))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 5.00000006e-8

   1. Initial program 65.1%

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

   3. Taylor expanded in dX.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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 50.0%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 54.2

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

   10. Applied rewrites 54.2%

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

   if 5.00000006e-8 < dX.v

   1. Initial program 60.9%

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

   3. Taylor expanded in dY.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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 52.5

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

   5. Applied rewrites 52.5%

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 55.6

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

   10. Applied rewrites 55.6%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.v \leq 11.199999809265137:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left(t\_1 + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_1, t\_0 + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0)) (t_1 (pow (* dX.w (floor d)) 2.0)))
   (if (<= dY.v 11.199999809265137)
     (log2
      (sqrt
       (fmax
        (+ (pow (* dX.v (floor h)) 2.0) (+ t_1 (pow (* dX.u (floor w)) 2.0)))
        t_0)))
     (log2
      (sqrt
       (fmax
        t_1
        (+
         t_0
         (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (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 = powf((dY_46_w * floorf(d)), 2.0f);
	float t_1 = powf((dX_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dY_46_v <= 11.199999809265137f) {
		tmp = log2f(sqrtf(fmaxf((powf((dX_46_v * floorf(h)), 2.0f) + (t_1 + powf((dX_46_u * floorf(w)), 2.0f))), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_1, (t_0 + (powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * 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(dY_46_w * floor(d)) ^ Float32(2.0)
	t_1 = Float32(dX_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(11.199999809265137))
		tmp = log2(sqrt(((Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_1 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) != Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_1 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0))))) ? t_0 : ((t_0 != t_0) ? Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_1 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))) : max(Float32((Float32(dX_46_v * floor(h)) ^ Float32(2.0)) + Float32(t_1 + (Float32(dX_46_u * floor(w)) ^ Float32(2.0)))), t_0)))));
	else
		tmp = log2(sqrt(((t_1 != t_1) ? Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) : ((Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))))) ? t_1 : max(t_1, Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (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 = (dY_46_w * floor(d)) ^ single(2.0);
	t_1 = (dX_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_v <= single(11.199999809265137))
		tmp = log2(sqrt(max((((dX_46_v * floor(h)) ^ single(2.0)) + (t_1 + ((dX_46_u * floor(w)) ^ single(2.0)))), t_0)));
	else
		tmp = log2(sqrt(max(t_1, (t_0 + (((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 11.1999998

   1. Initial program 63.5%

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

   3. Taylor expanded in dY.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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 53.2

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

   5. Applied rewrites 53.2%

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

   7. Applied rewrites 53.2%

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

   if 11.1999998 < dY.v

   1. Initial program 64.9%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 52.3

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

   5. Applied rewrites 52.3%

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 58.1

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

   10. Applied rewrites 58.1%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 11.199999809265137:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + \left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right), {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 150000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, t\_0 + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0)))
   (if (<= dX.v 150000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* dX.w (floor d)) 2.0)
        (+
         t_0
         (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 2.0))))))
     (log2 (sqrt (fmax (* (* (pow (floor h) 2.0) dX.v) dX.v) 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((dY_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_v <= 150000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_w * floorf(d)), 2.0f), (t_0 + (powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), 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 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(150000000.0))
		tmp = log2(sqrt((((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) ? Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) : ((Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))))) ? (Float32(dX_46_w * floor(d)) ^ Float32(2.0)) : max((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) : max(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), 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 = (dY_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(150000000.0))
		tmp = log2(sqrt(max(((dX_46_w * floor(d)) ^ single(2.0)), (t_0 + (((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.5e8

   1. Initial program 65.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 54.1

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

   10. Applied rewrites 54.1%

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

   if 1.5e8 < dX.v

   1. Initial program 52.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 49.9

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

   10. Applied rewrites 49.9%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 150000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 150000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, t\_0 + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0)))
   (if (<= dX.v 150000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* dX.u (floor w)) 2.0)
        (+
         t_0
         (+ (pow (* dY.u (floor w)) 2.0) (pow (* dY.v (floor h)) 2.0))))))
     (log2 (sqrt (fmax (* (* (pow (floor h) 2.0) dX.v) dX.v) 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((dY_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_v <= 150000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_u * floorf(w)), 2.0f), (t_0 + (powf((dY_46_u * floorf(w)), 2.0f) + powf((dY_46_v * floorf(h)), 2.0f))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), 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 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(150000000.0))
		tmp = log2(sqrt((((Float32(dX_46_u * floor(w)) ^ Float32(2.0)) != (Float32(dX_46_u * floor(w)) ^ Float32(2.0))) ? Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) : ((Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))))) ? (Float32(dX_46_u * floor(w)) ^ Float32(2.0)) : max((Float32(dX_46_u * floor(w)) ^ Float32(2.0)), Float32(t_0 + Float32((Float32(dY_46_u * floor(w)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))))))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) : max(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), 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 = (dY_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(150000000.0))
		tmp = log2(sqrt(max(((dX_46_u * floor(w)) ^ single(2.0)), (t_0 + (((dY_46_u * floor(w)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0)))))));
	else
		tmp = log2(sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.5e8

   1. Initial program 65.8%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   if 1.5e8 < dX.v

   1. Initial program 52.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 50.5%

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

   8. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 49.9

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

   10. Applied rewrites 49.9%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 150000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 0.10000000149011612:\\ \;\;\;\;\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, t\_0\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0)))
   (if (<= dX.v 0.10000000149011612)
     (log2 (pow (pow (fmax (pow (* dX.w (floor d)) 2.0) t_0) 0.25) 2.0))
     (log2 (sqrt (fmax (* (* (pow (floor h) 2.0) dX.v) dX.v) 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((dY_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_v <= 0.10000000149011612f) {
		tmp = log2f(powf(powf(fmaxf(powf((dX_46_w * floorf(d)), 2.0f), t_0), 0.25f), 2.0f));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), 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 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(0.10000000149011612))
		tmp = log2((((((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(dX_46_w * floor(d)) ^ Float32(2.0)) : max((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), t_0))) ^ Float32(0.25)) ^ Float32(2.0)));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) : max(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), 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 = (dY_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(0.10000000149011612))
		tmp = log2(((max(((dX_46_w * floor(d)) ^ single(2.0)), t_0) ^ single(0.25)) ^ single(2.0)));
	else
		tmp = log2(sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.100000001

   1. Initial program 65.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 37.8

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

   10. Applied rewrites 37.8%

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

       1. lift-sqrt.f32 N/A

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

       2. pow1/2 N/A

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

       3. sqr-pow N/A

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

   12. Applied rewrites 37.8%

       \[\leadsto \log_{2} \color{blue}{\left({\left({\left(\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)\right)}^{0.25}\right)}^{2}\right)} \]

   if 0.100000001 < dX.v

   1. Initial program 59.7%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 50.6%

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

   8. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 45.6

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

   10. Applied rewrites 45.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.10000000149011612:\\ \;\;\;\;\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 0.10000000149011612:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.w (floor d)) 2.0)))
   (if (<= dX.v 0.10000000149011612)
     (log2 (sqrt (fmax (pow (* dX.w (floor d)) 2.0) t_0)))
     (log2 (sqrt (fmax (* (* (pow (floor h) 2.0) dX.v) dX.v) 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((dY_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_v <= 0.10000000149011612f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_w * floorf(d)), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), 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 = Float32(dY_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(0.10000000149011612))
		tmp = log2(sqrt((((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(dX_46_w * floor(d)) ^ Float32(2.0)) : max((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) : max(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), 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 = (dY_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(0.10000000149011612))
		tmp = log2(sqrt(max(((dX_46_w * floor(d)) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.100000001

   1. Initial program 65.0%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 51.9

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

   5. Applied rewrites 51.9%

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 37.8

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

   10. Applied rewrites 37.8%

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

   if 0.100000001 < dX.v

   1. Initial program 59.7%

      \[\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloor w\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dX.u\right) + \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloor w\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot dY.u\right) + \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)}\right) \]
   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 50.6

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

   5. Applied rewrites 50.6%

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

   7. Applied rewrites 50.6%

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

   8. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 45.6

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

   10. Applied rewrites 45.6%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.10000000149011612:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dY.w \leq 65000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0)))
   (if (<= dY.w 65000.0)
     (log2 (sqrt (fmax t_0 (pow (* dY.v (floor h)) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (* dY.w (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((dX_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dY_46_w <= 65000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((dY_46_v * floorf(h)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((dY_46_w * 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(dX_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(65000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(dY_46_v * floor(h)) ^ Float32(2.0)) : (((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) != (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(dY_46_w * floor(d)) ^ Float32(2.0)) : (((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dY_46_w * floor(d)) ^ Float32(2.0))) ? t_0 : max(t_0, (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 = (dX_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_w <= single(65000.0))
		tmp = log2(sqrt(max(t_0, ((dY_46_v * floor(h)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((dY_46_w * floor(d)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 65000

   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.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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 52.6

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

   5. Applied rewrites 52.6%

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

   7. Applied rewrites 52.6%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 34.4

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

   10. Applied rewrites 34.4%

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

   11. Taylor expanded in dY.v around inf

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

       1. unpow2 N/A

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

       2. unpow2 N/A

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

       3. unswap-sqr N/A

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

       4. unpow2 N/A

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

       5. lower-pow.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lower-floor.f32 38.2

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

   13. Applied rewrites 38.2%

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

   if 65000 < 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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 47.9%

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

   8. Taylor expanded in dX.w around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-floor.f32 38.6

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

   10. Applied rewrites 38.6%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 65000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right) \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* dX.w (floor d)) 2.0) (pow (* dY.v (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) {
	return log2f(sqrtf(fmaxf(powf((dX_46_w * floorf(d)), 2.0f), powf((dY_46_v * floorf(h)), 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(dX_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) ? (Float32(dY_46_v * floor(h)) ^ Float32(2.0)) : (((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) != (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) ? (Float32(dX_46_w * floor(d)) ^ Float32(2.0)) : max((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), (Float32(dY_46_v * floor(h)) ^ 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(((dX_46_w * floor(d)) ^ single(2.0)), ((dY_46_v * floor(h)) ^ single(2.0)))));
end

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in dY.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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-floor.f32 51.6

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

5. Applied rewrites 51.6%

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

7. Applied rewrites 51.6%

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

8. Taylor expanded in dX.w around inf

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. unswap-sqr N/A

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

   4. unpow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-floor.f32 35.3

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

10. Applied rewrites 35.3%

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

11. Taylor expanded in dY.v around inf

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

    1. unpow2 N/A

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

    2. unpow2 N/A

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

    3. unswap-sqr N/A

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

    4. unpow2 N/A

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

    5. lower-pow.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-floor.f32 35.1

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

13. Applied rewrites 35.1%

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

14. Final simplification 35.1%

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

Reproduce

herbie shell --seed 2024250 
(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)))))))