Isotropic LOD (LOD)

Percentage Accurate: 68.0% → 71.3%

Time: 11.5s

Alternatives: 17

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(fmax(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))
end

MATLAB

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

Initial Program: 68.0% 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(fmax(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: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, \left(t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4\right)}\right)\\ \mathbf{if}\;t\_6 \leq 63.900001525878906:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right), \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor d) dY.w))
        (t_5 (* (floor d) dX.w))
        (t_6
         (log2
          (sqrt
           (fmax
            (+ (+ (* t_0 t_0) (* t_3 t_3)) (* t_5 t_5))
            (+ (+ (* t_1 t_1) (* t_2 t_2)) (* t_4 t_4)))))))
   (if (<= t_6 63.900001525878906)
     t_6
     (log2
      (sqrt
       (fmax
        (* (* (floor w) (floor w)) (* dX.u dX.u))
        (* (* (* dY.v (floor h)) dY.v) (floor h))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_3 * t_3)) + (t_5 * t_5)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	float tmp;
	if (t_6 <= 63.900001525878906f) {
		tmp = t_6;
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(w) * floorf(w)) * (dX_46_u * dX_46_u)), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)))))
	tmp = Float32(0.0)
	if (t_6 <= Float32(63.900001525878906))
		tmp = t_6;
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(w) * floor(w)) * Float32(dX_46_u * dX_46_u)), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(w) * dY_46_u;
	t_2 = floor(h) * dY_46_v;
	t_3 = floor(h) * dX_46_v;
	t_4 = floor(d) * dY_46_w;
	t_5 = floor(d) * dX_46_w;
	t_6 = log2(sqrt(max((((t_0 * t_0) + (t_3 * t_3)) + (t_5 * t_5)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	tmp = single(0.0);
	if (t_6 <= single(63.900001525878906))
		tmp = t_6;
	else
		tmp = log2(sqrt(max(((floor(w) * floor(w)) * (dX_46_u * dX_46_u)), (((dY_46_v * floor(h)) * dY_46_v) * floor(h)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (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)))))) < 63.9000015

   1. Initial program 100.0%

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

   if 63.9000015 < (log2.f32 (sqrt.f32 (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 7.3%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 13.3

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

   4. Applied rewrites 13.3%

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

   5. Taylor expanded in dY.v around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 16.7%

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

Alternative 2: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_5 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_6 := \left\lfloor w\right\rfloor \cdot dX.u\\ \mathbf{if}\;dX.w \leq 2000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, t\_4 \cdot \left(dX.v \cdot dX.v\right)\right), \left(t\_5 \cdot t\_5 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_6 \cdot t\_6 + t\_0 \cdot t\_0\right) + t\_2 \cdot t\_2, \left(dY.v \cdot dY.v\right) \cdot t\_4\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor d) dX.w))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor h) (floor h)))
        (t_5 (* (floor w) dY.u))
        (t_6 (* (floor w) dX.u)))
   (if (<= dX.w 2000000.0)
     (log2
      (sqrt
       (fmax
        (fma (* (* (floor w) (floor w)) dX.u) dX.u (* t_4 (* dX.v dX.v)))
        (+ (+ (* t_5 t_5) (* t_1 t_1)) (* t_3 t_3)))))
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_6 t_6) (* t_0 t_0)) (* t_2 t_2))
        (* (* dY.v dY.v) t_4)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(d) * dX_46_w;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(h) * floorf(h);
	float t_5 = floorf(w) * dY_46_u;
	float t_6 = floorf(w) * dX_46_u;
	float tmp;
	if (dX_46_w <= 2000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(w) * floorf(w)) * dX_46_u), dX_46_u, (t_4 * (dX_46_v * dX_46_v))), (((t_5 * t_5) + (t_1 * t_1)) + (t_3 * t_3)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_6 * t_6) + (t_0 * t_0)) + (t_2 * t_2)), ((dY_46_v * dY_46_v) * t_4))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(d) * dX_46_w)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(h) * floor(h))
	t_5 = Float32(floor(w) * dY_46_u)
	t_6 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(2000000.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(w) * floor(w)) * dX_46_u), dX_46_u, Float32(t_4 * Float32(dX_46_v * dX_46_v))), Float32(Float32(Float32(t_5 * t_5) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_6 * t_6) + Float32(t_0 * t_0)) + Float32(t_2 * t_2)), Float32(Float32(dY_46_v * dY_46_v) * t_4))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 2e6

   1. Initial program 69.8%

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

   2. Taylor expanded in dX.w around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lift-floor.f32 N/A

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

      9. lift-floor.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. lift-floor.f32 N/A

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

      15. lift-floor.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 64.7

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

   4. Applied rewrites 64.7%

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

   if 2e6 < dX.w

   1. Initial program 59.9%

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

   2. Taylor expanded in dY.v around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 54.5

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

   4. Applied rewrites 54.5%

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

Alternative 3: 56.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := \left\lfloor w\right\rfloor \cdot dY.u\\ \mathbf{if}\;dY.w \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, \left(dY.u \cdot dY.u\right) \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_2 \cdot \left(dX.u \cdot dX.u\right), \left(t\_6 \cdot t\_6 + t\_4 \cdot t\_4\right) + t\_0 \cdot t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w))
        (t_1 (* (floor w) dX.u))
        (t_2 (* (floor w) (floor w)))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor h) dY.v))
        (t_5 (* (floor d) dX.w))
        (t_6 (* (floor w) dY.u)))
   (if (<= dY.w 100.0)
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_1 t_1) (* t_3 t_3)) (* t_5 t_5))
        (* (* dY.u dY.u) t_2))))
     (log2
      (sqrt
       (fmax
        (* t_2 (* dX.u dX.u))
        (+ (+ (* t_6 t_6) (* t_4 t_4)) (* t_0 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 = floorf(d) * dY_46_w;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = floorf(w) * floorf(w);
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(h) * dY_46_v;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = floorf(w) * dY_46_u;
	float tmp;
	if (dY_46_w <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf((((t_1 * t_1) + (t_3 * t_3)) + (t_5 * t_5)), ((dY_46_u * dY_46_u) * t_2))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_2 * (dX_46_u * dX_46_u)), (((t_6 * t_6) + (t_4 * t_4)) + (t_0 * 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(floor(d) * dY_46_w)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = Float32(floor(w) * floor(w))
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(h) * dY_46_v)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(100.0))
		tmp = log2(sqrt(fmax(Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(Float32(dY_46_u * dY_46_u) * t_2))));
	else
		tmp = log2(sqrt(fmax(Float32(t_2 * Float32(dX_46_u * dX_46_u)), Float32(Float32(Float32(t_6 * t_6) + Float32(t_4 * t_4)) + Float32(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	t_1 = floor(w) * dX_46_u;
	t_2 = floor(w) * floor(w);
	t_3 = floor(h) * dX_46_v;
	t_4 = floor(h) * dY_46_v;
	t_5 = floor(d) * dX_46_w;
	t_6 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if (dY_46_w <= single(100.0))
		tmp = log2(sqrt(max((((t_1 * t_1) + (t_3 * t_3)) + (t_5 * t_5)), ((dY_46_u * dY_46_u) * t_2))));
	else
		tmp = log2(sqrt(max((t_2 * (dX_46_u * dX_46_u)), (((t_6 * t_6) + (t_4 * t_4)) + (t_0 * t_0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 100

   1. Initial program 69.2%

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

   2. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 56.9

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

   4. Applied rewrites 56.9%

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

   if 100 < dY.w

   1. Initial program 64.0%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.3

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

   4. Applied rewrites 56.3%

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

Alternative 4: 56.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ \mathbf{if}\;dY.w \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot dX.v, dX.v, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w\right) \cdot \left\lfloor d\right\rfloor \right)\right), \left(dY.u \cdot dY.u\right) \cdot t\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_2 \cdot \left(dX.u \cdot dX.u\right), \left(t\_3 \cdot t\_3 + t\_0 \cdot t\_0\right) + t\_1 \cdot t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor d) dY.w))
        (t_2 (* (floor w) (floor w)))
        (t_3 (* (floor w) dY.u)))
   (if (<= dY.w 100.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* (* (floor h) (floor h)) dX.v)
         dX.v
         (fma
          (* (* dX.u (floor w)) dX.u)
          (floor w)
          (* (* (* dX.w (floor d)) dX.w) (floor d))))
        (* (* dY.u dY.u) t_2))))
     (log2
      (sqrt
       (fmax
        (* t_2 (* dX.u dX.u))
        (+ (+ (* 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 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = floorf(w) * floorf(w);
	float t_3 = floorf(w) * dY_46_u;
	float tmp;
	if (dY_46_w <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_w * floorf(d)) * dX_46_w) * floorf(d)))), ((dY_46_u * dY_46_u) * t_2))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_2 * (dX_46_u * dX_46_u)), (((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(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(floor(w) * floor(w))
	t_3 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(100.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_w * floor(d)) * dX_46_w) * floor(d)))), Float32(Float32(dY_46_u * dY_46_u) * t_2))));
	else
		tmp = log2(sqrt(fmax(Float32(t_2 * Float32(dX_46_u * dX_46_u)), Float32(Float32(Float32(t_3 * t_3) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 100

   1. Initial program 69.2%

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

   2. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 56.9

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

   4. Applied rewrites 56.9%

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

   6. Applied rewrites 56.9%

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

   if 100 < dY.w

   1. Initial program 64.0%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.3

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

   4. Applied rewrites 56.3%

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

Alternative 5: 56.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor d) dY.w))
        (t_2 (* (floor w) dY.u)))
   (if (<= dY.w 2.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* (* (floor h) (floor h)) dX.v)
         dX.v
         (fma
          (* (* dX.u (floor w)) dX.u)
          (floor w)
          (* (* (* dX.w (floor d)) dX.w) (floor d))))
        (* (* dY.u dY.u) (* (floor w) (floor w))))))
     (log2
      (sqrt
       (fmax
        (* (* (floor d) (floor d)) (* dX.w dX.w))
        (+ (+ (* t_2 t_2) (* 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 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = floorf(w) * dY_46_u;
	float tmp;
	if (dY_46_w <= 2.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(h) * floorf(h)) * dX_46_v), dX_46_v, fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_w * floorf(d)) * dX_46_w) * floorf(d)))), ((dY_46_u * dY_46_u) * (floorf(w) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w)), (((t_2 * t_2) + (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(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(2.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(h) * floor(h)) * dX_46_v), dX_46_v, fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_w * floor(d)) * dX_46_w) * floor(d)))), Float32(Float32(dY_46_u * dY_46_u) * Float32(floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)), Float32(Float32(Float32(t_2 * t_2) + Float32(t_0 * t_0)) + Float32(t_1 * t_1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 2

   1. Initial program 69.0%

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

   2. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 56.7

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

   4. Applied rewrites 56.7%

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

   6. Applied rewrites 56.7%

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

   if 2 < dY.w

   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. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 55.5

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

   4. Applied rewrites 55.5%

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

Alternative 6: 55.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ \mathbf{if}\;dX.u \leq 20000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 \cdot \left(dX.v \cdot dX.v\right), \mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right), \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0 \cdot dX.v, dX.v, \mathsf{fma}\left(\left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, \left\lfloor w\right\rfloor , \left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot dX.w\right) \cdot \left\lfloor d\right\rfloor \right)\right), \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) (floor h))))
   (if (<= dX.u 20000000000.0)
     (log2
      (sqrt
       (fmax
        (* t_0 (* dX.v dX.v))
        (fma
         (floor h)
         (* (floor h) (* dY.v dY.v))
         (fma
          (* (* dY.w (floor d)) dY.w)
          (floor d)
          (* (* (* dY.u (floor w)) dY.u) (floor w)))))))
     (log2
      (sqrt
       (fmax
        (fma
         (* t_0 dX.v)
         dX.v
         (fma
          (* (* dX.u (floor w)) dX.u)
          (floor w)
          (* (* (* dX.w (floor d)) dX.w) (floor d))))
        (* (* dY.u dY.u) (* (floor w) (floor w)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * floorf(h);
	float tmp;
	if (dX_46_u <= 20000000000.0f) {
		tmp = log2f(sqrtf(fmaxf((t_0 * (dX_46_v * dX_46_v)), fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w)))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_0 * dX_46_v), dX_46_v, fmaf(((dX_46_u * floorf(w)) * dX_46_u), floorf(w), (((dX_46_w * floorf(d)) * dX_46_w) * floorf(d)))), ((dY_46_u * dY_46_u) * (floorf(w) * floorf(w))))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * floor(h))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(20000000000.0))
		tmp = log2(sqrt(fmax(Float32(t_0 * Float32(dX_46_v * dX_46_v)), fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w)))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_0 * dX_46_v), dX_46_v, fma(Float32(Float32(dX_46_u * floor(w)) * dX_46_u), floor(w), Float32(Float32(Float32(dX_46_w * floor(d)) * dX_46_w) * floor(d)))), Float32(Float32(dY_46_u * dY_46_u) * Float32(floor(w) * floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 2e10

   1. Initial program 70.1%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.7

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

   4. Applied rewrites 56.7%

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

   6. Applied rewrites 56.7%

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

   if 2e10 < dX.u

   1. Initial program 52.6%

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

   2. Taylor expanded in dY.u around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-floor.f32 50.2

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

   4. Applied rewrites 50.2%

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

   6. Applied rewrites 50.2%

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

Alternative 7: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w, \left\lfloor d\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\\ \mathbf{if}\;dX.u \leq 30000001024:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \mathsf{fma}\left(\left\lfloor h\right\rfloor , \left\lfloor h\right\rfloor \cdot \left(dY.v \cdot dY.v\right), t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right), t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (fma
          (* (* dY.w (floor d)) dY.w)
          (floor d)
          (* (* (* dY.u (floor w)) dY.u) (floor w)))))
   (if (<= dX.u 30000001024.0)
     (log2
      (sqrt
       (fmax
        (* (* (floor h) (floor h)) (* dX.v dX.v))
        (fma (floor h) (* (floor h) (* dY.v dY.v)) t_0))))
     (log2 (sqrt (fmax (* (* (floor w) (floor w)) (* dX.u dX.u)) t_0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = fmaf(((dY_46_w * floorf(d)) * dY_46_w), floorf(d), (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w)));
	float tmp;
	if (dX_46_u <= 30000001024.0f) {
		tmp = log2f(sqrtf(fmaxf(((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v)), fmaf(floorf(h), (floorf(h) * (dY_46_v * dY_46_v)), t_0))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(w) * floorf(w)) * (dX_46_u * dX_46_u)), t_0)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = fma(Float32(Float32(dY_46_w * floor(d)) * dY_46_w), floor(d), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w)))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(30000001024.0))
		tmp = log2(sqrt(fmax(Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)), fma(floor(h), Float32(floor(h) * Float32(dY_46_v * dY_46_v)), t_0))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(w) * floor(w)) * Float32(dX_46_u * dX_46_u)), t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 30000001000

   1. Initial program 70.1%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.6

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

   4. Applied rewrites 56.6%

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

   6. Applied rewrites 56.6%

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

   if 30000001000 < dX.u

   1. Initial program 52.4%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in dY.v around 0

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

   7. Applied rewrites 45.5%

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

Alternative 8: 48.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\\ \mathbf{if}\;dY.u \leq 32000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right), \mathsf{fma}\left(t\_0, \left\lfloor d\right\rfloor , \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \mathsf{fma}\left(t\_0, \left\lfloor d\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (* dY.w (floor d)) dY.w)))
   (if (<= dY.u 32000.0)
     (log2
      (sqrt
       (fmax
        (* (* (floor w) (floor w)) (* dX.u dX.u))
        (fma t_0 (floor d) (* (* (* dY.v (floor h)) dY.v) (floor h))))))
     (log2
      (sqrt
       (fmax
        (* (* (floor h) (floor h)) (* dX.v dX.v))
        (fma t_0 (floor d) (* (* (* dY.u (floor w)) dY.u) (floor w)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = (dY_46_w * floorf(d)) * dY_46_w;
	float tmp;
	if (dY_46_u <= 32000.0f) {
		tmp = log2f(sqrtf(fmaxf(((floorf(w) * floorf(w)) * (dX_46_u * dX_46_u)), fmaf(t_0, floorf(d), (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v)), fmaf(t_0, floorf(d), (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w))))));
	}
	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(Float32(dY_46_w * floor(d)) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(32000.0))
		tmp = log2(sqrt(fmax(Float32(Float32(floor(w) * floor(w)) * Float32(dX_46_u * dX_46_u)), fma(t_0, floor(d), Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)), fma(t_0, floor(d), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 32000

   1. Initial program 69.9%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.8

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in dY.u around 0

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

   7. Applied rewrites 47.9%

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

   if 32000 < dY.u

   1. Initial program 60.1%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in dY.v around 0

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

   7. Applied rewrites 50.6%

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

Alternative 9: 48.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\\ t_1 := \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\\ \mathbf{if}\;dY.u \leq 450:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot dX.u, dX.u, t\_0\right), t\_1 \cdot \left\lfloor d\right\rfloor \right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \mathsf{fma}\left(t\_1, \left\lfloor d\right\rfloor , \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (* (floor h) (floor h)) (* dX.v dX.v)))
        (t_1 (* (* dY.w (floor d)) dY.w)))
   (if (<= dY.u 450.0)
     (log2
      (sqrt
       (fmax
        (fma (* (* (floor w) (floor w)) dX.u) dX.u t_0)
        (* t_1 (floor d)))))
     (log2
      (sqrt
       (fmax
        t_0
        (fma t_1 (floor d) (* (* (* dY.u (floor w)) dY.u) (floor w)))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = (floorf(h) * floorf(h)) * (dX_46_v * dX_46_v);
	float t_1 = (dY_46_w * floorf(d)) * dY_46_w;
	float tmp;
	if (dY_46_u <= 450.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(((floorf(w) * floorf(w)) * dX_46_u), dX_46_u, t_0), (t_1 * floorf(d)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, fmaf(t_1, floorf(d), (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w))))));
	}
	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(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))
	t_1 = Float32(Float32(dY_46_w * floor(d)) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(450.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(floor(w) * floor(w)) * dX_46_u), dX_46_u, t_0), Float32(t_1 * floor(d)))));
	else
		tmp = log2(sqrt(fmax(t_0, fma(t_1, floor(d), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 450

   1. Initial program 69.9%

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

   2. Taylor expanded in dX.w around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lift-floor.f32 N/A

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

      9. lift-floor.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. lift-floor.f32 N/A

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

      15. lift-floor.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 47.6%

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

   if 450 < dY.u

   1. Initial program 61.3%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in dY.v around 0

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

   7. Applied rewrites 50.2%

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

Alternative 10: 48.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_3 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_6 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_4 \cdot t\_4\right) + t\_6 \cdot t\_6, \left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5\right)}\right) \leq 48:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), \mathsf{fma}\left(t\_2 \cdot dY.u, dY.u, \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_2 \cdot dX.u, dX.u, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right), \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor w) (floor w)))
        (t_3 (* (floor h) dY.v))
        (t_4 (* (floor h) dX.v))
        (t_5 (* (floor d) dY.w))
        (t_6 (* (floor d) dX.w)))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_0 t_0) (* t_4 t_4)) (* t_6 t_6))
           (+ (+ (* t_1 t_1) (* t_3 t_3)) (* t_5 t_5)))))
        48.0)
     (log2
      (sqrt
       (fmax
        (* (* (floor d) (floor d)) (* dX.w dX.w))
        (fma (* t_2 dY.u) dY.u (* (* (* dY.v (floor h)) dY.v) (floor h))))))
     (log2
      (sqrt
       (fmax
        (fma (* t_2 dX.u) dX.u (* (* (floor h) (floor h)) (* dX.v dX.v)))
        (* (* (* dY.w (floor d)) dY.w) (floor d))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(w) * floorf(w);
	float t_3 = floorf(h) * dY_46_v;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w;
	float t_6 = floorf(d) * dX_46_w;
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_4 * t_4)) + (t_6 * t_6)), (((t_1 * t_1) + (t_3 * t_3)) + (t_5 * t_5))))) <= 48.0f) {
		tmp = log2f(sqrtf(fmaxf(((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w)), fmaf((t_2 * dY_46_u), dY_46_u, (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((t_2 * dX_46_u), dX_46_u, ((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v))), (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(w) * floor(w))
	t_3 = Float32(floor(h) * dY_46_v)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w)
	t_6 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)) + Float32(t_6 * t_6)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5))))) <= Float32(48.0))
		tmp = log2(sqrt(fmax(Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)), fma(Float32(t_2 * dY_46_u), dY_46_u, Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(t_2 * dX_46_u), dX_46_u, Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))), Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (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)))))) < 48

   1. Initial program 99.9%

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

   2. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in dY.w around 0

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

      1. pow2 N/A

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

      2. lift-floor.f32 N/A

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

      3. lift-floor.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift-floor.f32 N/A

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

      8. lift-floor.f32 N/A

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

      9. pow2 N/A

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

      10. lift-floor.f32 N/A

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

      11. lift-floor.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. associate-*r* N/A

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

   7. Applied rewrites 59.2%

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

   if 48 < (log2.f32 (sqrt.f32 (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 55.5%

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

   2. Taylor expanded in dX.w around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lift-floor.f32 N/A

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

      9. lift-floor.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. lift-floor.f32 N/A

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

      15. lift-floor.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 50.8

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

   4. Applied rewrites 50.8%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 39.9%

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

Alternative 11: 47.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_1 := \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dX.u \leq 80:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, dX.w \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right)\right), t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_0 \cdot \left(dX.w \cdot dX.w\right)\right), t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) (floor d)))
        (t_1 (* (* (* dY.w (floor d)) dY.w) (floor d))))
   (if (<= dX.u 80.0)
     (log2
      (sqrt
       (fmax
        (fma t_0 (* dX.w dX.w) (* (* (floor h) (floor h)) (* dX.v dX.v)))
        t_1)))
     (log2
      (sqrt
       (fmax
        (fma (* dX.u dX.u) (* (floor w) (floor w)) (* t_0 (* dX.w dX.w)))
        t_1))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * floorf(d);
	float t_1 = ((dY_46_w * floorf(d)) * dY_46_w) * floorf(d);
	float tmp;
	if (dX_46_u <= 80.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, (dX_46_w * dX_46_w), ((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v))), t_1)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), (floorf(w) * floorf(w)), (t_0 * (dX_46_w * dX_46_w))), t_1)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * floor(d))
	t_1 = Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(80.0))
		tmp = log2(sqrt(fmax(fma(t_0, Float32(dX_46_w * dX_46_w), Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v))), t_1)));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), Float32(floor(w) * floor(w)), Float32(t_0 * Float32(dX_46_w * dX_46_w))), t_1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 80

   1. Initial program 69.8%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 54.0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 32.8%

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

   8. Taylor expanded in dX.u around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lift-floor.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. pow2 N/A

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

      12. lift-floor.f32 N/A

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

      13. lift-floor.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 47.6

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

   10. Applied rewrites 47.6%

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

   if 80 < dX.u

   1. Initial program 61.8%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 52.7

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 45.0%

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

   8. Taylor expanded in dX.v around 0

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

      1. pow2 N/A

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

      2. lower-fma.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-floor.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lift-floor.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. pow2 N/A

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

      13. lift-*.f32 49.6

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

   10. Applied rewrites 49.6%

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

Alternative 12: 45.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dX.v \leq 100000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right)\right), t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (* (* dY.w (floor d)) dY.w) (floor d))))
   (if (<= dX.v 100000.0)
     (log2
      (sqrt
       (fmax
        (fma
         (* dX.u dX.u)
         (* (floor w) (floor w))
         (* (* (floor d) (floor d)) (* dX.w dX.w)))
        t_0)))
     (log2 (sqrt (fmax (* (* (floor h) (floor h)) (* 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 = ((dY_46_w * floorf(d)) * dY_46_w) * floorf(d);
	float tmp;
	if (dX_46_v <= 100000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), (floorf(w) * floorf(w)), ((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w))), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(h) * floorf(h)) * (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(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d))
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(100000.0))
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), Float32(floor(w) * floor(w)), Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))), t_0)));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)), t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1e5

   1. Initial program 69.6%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.9

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in dX.v around 0

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

      1. pow2 N/A

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

      2. lower-fma.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-floor.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lift-floor.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. pow2 N/A

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

      13. lift-*.f32 47.5

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

   10. Applied rewrites 47.5%

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

   if 1e5 < dX.v

   1. Initial program 61.3%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 47.6%

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

Alternative 13: 39.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 11:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot dX.u\right), \left(\left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot dY.u\right) \cdot \left\lfloor w\right\rfloor \right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 11.0)
   (log2
    (sqrt
     (fmax
      (* (* (floor w) (floor w)) (* dX.u dX.u))
      (* (* (* dY.u (floor w)) dY.u) (floor w)))))
   (log2
    (sqrt
     (fmax
      (* (* (floor h) (floor h)) (* dX.v dX.v))
      (* (* (* dY.w (floor d)) dY.w) (floor d)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 11.0f) {
		tmp = log2f(sqrtf(fmaxf(((floorf(w) * floorf(w)) * (dX_46_u * dX_46_u)), (((dY_46_u * floorf(w)) * dY_46_u) * floorf(w)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v)), (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(11.0))
		tmp = log2(sqrt(fmax(Float32(Float32(floor(w) * floor(w)) * Float32(dX_46_u * dX_46_u)), Float32(Float32(Float32(dY_46_u * floor(w)) * dY_46_u) * floor(w)))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)), Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(11.0))
		tmp = log2(sqrt(max(((floor(w) * floor(w)) * (dX_46_u * dX_46_u)), (((dY_46_u * floor(w)) * dY_46_u) * floor(w)))));
	else
		tmp = log2(sqrt(max(((floor(h) * floor(h)) * (dX_46_v * dX_46_v)), (((dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 11

   1. Initial program 69.4%

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

   2. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f32 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-log.f32 40.3

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

      7. lift-*.f32 N/A

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

      8. lift-floor.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lift-floor.f32 40.3

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

   6. Applied rewrites 40.3%

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

   7. Taylor expanded in dY.u around inf

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

      1. exp-to-pow N/A

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

      2. pow-prod-down N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. unpow2 N/A

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

      6. swap-sqr N/A

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

      7. unpow2 N/A

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

      8. lift-floor.f32 N/A

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

      9. pow2 N/A

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

      10. swap-sqr N/A

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

      11. lift-floor.f32 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f32 N/A

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

   9. Applied rewrites 37.1%

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

   if 11 < dX.v

   1. Initial program 63.6%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 45.6%

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

Alternative 14: 39.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 18:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right), \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot dX.v\right), \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 18.0)
   (log2
    (sqrt
     (fmax
      (* (* (floor d) (floor d)) (* dX.w dX.w))
      (* (* dY.u dY.u) (* (floor w) (floor w))))))
   (log2
    (sqrt
     (fmax
      (* (* (floor h) (floor h)) (* dX.v dX.v))
      (* (* (* dY.w (floor d)) dY.w) (floor d)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 18.0f) {
		tmp = log2f(sqrtf(fmaxf(((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w)), ((dY_46_u * dY_46_u) * (floorf(w) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((floorf(h) * floorf(h)) * (dX_46_v * dX_46_v)), (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(18.0))
		tmp = log2(sqrt(fmax(Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)), Float32(Float32(dY_46_u * dY_46_u) * Float32(floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(Float32(Float32(floor(h) * floor(h)) * Float32(dX_46_v * dX_46_v)), Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(18.0))
		tmp = log2(sqrt(max(((floor(d) * floor(d)) * (dX_46_w * dX_46_w)), ((dY_46_u * dY_46_u) * (floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(max(((floor(h) * floor(h)) * (dX_46_v * dX_46_v)), (((dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 18

   1. Initial program 69.5%

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

   2. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 56.6

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

   4. Applied rewrites 56.6%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in dY.u around inf

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

      1. pow2 N/A

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

      2. lift-floor.f32 N/A

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

      3. lift-floor.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 37.1

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

   10. Applied rewrites 37.1%

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

   if 18 < dX.v

   1. Initial program 63.2%

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

   2. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.8

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 45.7%

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

Alternative 15: 39.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right)\\ \mathbf{if}\;dY.w \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \left(\left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot dY.w\right) \cdot \left\lfloor d\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (* (floor d) (floor d)) (* dX.w dX.w))))
   (if (<= dY.w 20.0)
     (log2 (sqrt (fmax t_0 (* (* dY.u dY.u) (* (floor w) (floor w))))))
     (log2 (sqrt (fmax t_0 (* (* (* dY.w (floor d)) dY.w) (floor d))))))))

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(d) * floorf(d)) * (dX_46_w * dX_46_w);
	float tmp;
	if (dY_46_w <= 20.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, ((dY_46_u * dY_46_u) * (floorf(w) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, (((dY_46_w * floorf(d)) * dY_46_w) * floorf(d)))));
	}
	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(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(20.0))
		tmp = log2(sqrt(fmax(t_0, Float32(Float32(dY_46_u * dY_46_u) * Float32(floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(t_0, Float32(Float32(Float32(dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (floor(d) * floor(d)) * (dX_46_w * dX_46_w);
	tmp = single(0.0);
	if (dY_46_w <= single(20.0))
		tmp = log2(sqrt(max(t_0, ((dY_46_u * dY_46_u) * (floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(max(t_0, (((dY_46_w * floor(d)) * dY_46_w) * floor(d)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 20

   1. Initial program 69.2%

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

   2. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.0

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 32.5%

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

   8. Taylor expanded in dY.u around inf

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

      1. pow2 N/A

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

      2. lift-floor.f32 N/A

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

      3. lift-floor.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 37.2

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

   10. Applied rewrites 37.2%

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

   if 20 < dY.w

   1. Initial program 64.3%

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

   2. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 55.6

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

   4. Applied rewrites 55.6%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 45.0%

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

Alternative 16: 39.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot dX.w\right)\\ \mathbf{if}\;dY.v \leq 1:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \left(dY.u \cdot dY.u\right) \cdot \left(\left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot dY.v\right) \cdot \left\lfloor h\right\rfloor \right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (* (floor d) (floor d)) (* dX.w dX.w))))
   (if (<= dY.v 1.0)
     (log2 (sqrt (fmax t_0 (* (* dY.u dY.u) (* (floor w) (floor w))))))
     (log2 (sqrt (fmax t_0 (* (* (* dY.v (floor h)) dY.v) (floor h))))))))

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(d) * floorf(d)) * (dX_46_w * dX_46_w);
	float tmp;
	if (dY_46_v <= 1.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, ((dY_46_u * dY_46_u) * (floorf(w) * floorf(w))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, (((dY_46_v * floorf(h)) * dY_46_v) * floorf(h)))));
	}
	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(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w))
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(1.0))
		tmp = log2(sqrt(fmax(t_0, Float32(Float32(dY_46_u * dY_46_u) * Float32(floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(fmax(t_0, Float32(Float32(Float32(dY_46_v * floor(h)) * dY_46_v) * floor(h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (floor(d) * floor(d)) * (dX_46_w * dX_46_w);
	tmp = single(0.0);
	if (dY_46_v <= single(1.0))
		tmp = log2(sqrt(max(t_0, ((dY_46_u * dY_46_u) * (floor(w) * floor(w))))));
	else
		tmp = log2(sqrt(max(t_0, (((dY_46_v * floor(h)) * dY_46_v) * floor(h)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 1

   1. Initial program 69.6%

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

   2. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 53.2

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in dY.w around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in dY.u around inf

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

      1. pow2 N/A

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

      2. lift-floor.f32 N/A

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

      3. lift-floor.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 37.2

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

   10. Applied rewrites 37.2%

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

   if 1 < dY.v

   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. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-floor.f32 N/A

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

      6. lift-floor.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in dY.v around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-floor.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-*.f32 N/A

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

   7. Applied rewrites 44.7%

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

Alternative 17: 35.8% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (* (* (floor d) (floor d)) (* dX.w dX.w))
    (* (* dY.u dY.u) (* (floor w) (floor w)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(((floorf(d) * floorf(d)) * (dX_46_w * dX_46_w)), ((dY_46_u * dY_46_u) * (floorf(w) * floorf(w))))));
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(fmax(Float32(Float32(floor(d) * floor(d)) * Float32(dX_46_w * dX_46_w)), Float32(Float32(dY_46_u * dY_46_u) * Float32(floor(w) * floor(w))))))
end

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((floor(d) * floor(d)) * (dX_46_w * dX_46_w)), ((dY_46_u * dY_46_u) * (floor(w) * floor(w))))));
end

Derivation

1. Initial program 68.0%

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

2. Taylor expanded in dX.w around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-floor.f32 N/A

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

   6. lift-floor.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 53.6

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

4. Applied rewrites 53.6%

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

5. Taylor expanded in dY.w around inf

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

   1. pow2 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-floor.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

7. Applied rewrites 35.5%

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

8. Taylor expanded in dY.u around inf

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

   1. pow2 N/A

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

   2. lift-floor.f32 N/A

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

   3. lift-floor.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 35.8

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

10. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2025120 
(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)))))))