Isotropic LOD (LOD)

Percentage Accurate: 67.5% → 70.7%

Time: 27.2s

Alternatives: 21

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right) \leq 3.0000000054977558 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_4, \mathsf{hypot}\left(t\_5, t\_2\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_3, \mathsf{hypot}\left(t\_0, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_2}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (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)))
   (if (<=
        (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)))
        3.0000000054977558e+38)
     (log2
      (sqrt
       (fmax
        (pow (hypot t_4 (hypot t_5 t_2)) 2.0)
        (pow (hypot t_3 (hypot t_0 t_1)) 2.0))))
     (log2
      (sqrt (fmax (pow t_2 2.0) (* (pow (floor h) 2.0) (pow dY.v 2.0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	float tmp;
	if (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))) <= 3.0000000054977558e+38f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_4, hypotf(t_5, t_2)), 2.0f), powf(hypotf(t_3, hypotf(t_0, t_1)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_2, 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (((Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) != Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)) : max(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3))))) <= Float32(3.0000000054977558e+38))
		tmp = log2(sqrt((((hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)) != (hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0))) ? (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)) : (((hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)) != (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0))) ? (hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)) : max((hypot(t_4, hypot(t_5, t_2)) ^ Float32(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((t_2 ^ Float32(2.0)) != (t_2 ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (t_2 ^ Float32(2.0)) : max((t_2 ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dY_46_u;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dY_46_w;
	t_4 = floor(d) * dX_46_w;
	t_5 = floor(w) * dX_46_u;
	tmp = single(0.0);
	if (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))) <= single(3.0000000054977558e+38))
		tmp = log2(sqrt(max((hypot(t_4, hypot(t_5, t_2)) ^ single(2.0)), (hypot(t_3, hypot(t_0, t_1)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_2 ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))) < 3.00000001e38

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0 100.0%

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

   5. Simplified 100.0%

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

   if 3.00000001e38 < (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 7.6%

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

   3. Taylor expanded in w around 0 7.6%

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

   5. Simplified 7.6%

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

   6. Taylor expanded in dY.v around inf 14.2%

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

      1. *-commutative 14.2%

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

   8. Simplified 14.2%

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

   9. Taylor expanded in dX.v around inf 19.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right) \leq 3.0000000054977558 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_1 := \left\lfloord\right\rfloor \cdot dX.w\\ t_2 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dY.u \leq 10:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_1, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_2, t\_0\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.v}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2}, {t\_1}^{2}\right), {\left(\mathsf{hypot}\left(t\_2, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, t\_0\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v))
        (t_1 (* (floor d) dX.w))
        (t_2 (* (floor d) dY.w)))
   (if (<= dY.u 10.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot t_1 (hypot (* (floor w) dX.u) (* (floor h) dX.v))) 2.0)
        (pow (hypot t_2 t_0) 2.0))))
     (log2
      (sqrt
       (fmax
        (fma (pow dX.v 2.0) (pow (floor h) 2.0) (pow t_1 2.0))
        (pow (hypot t_2 (hypot (* (floor w) dY.u) t_0)) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dX_46_w;
	float t_2 = floorf(d) * dY_46_w;
	float tmp;
	if (dY_46_u <= 10.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_1, hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(hypotf(t_2, t_0), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(powf(dX_46_v, 2.0f), powf(floorf(h), 2.0f), powf(t_1, 2.0f)), powf(hypotf(t_2, hypotf((floorf(w) * dY_46_u), t_0)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dX_46_w)
	t_2 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(10.0))
		tmp = log2(sqrt((((hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (hypot(t_2, t_0) ^ Float32(2.0)) : (((hypot(t_2, t_0) ^ Float32(2.0)) != (hypot(t_2, t_0) ^ Float32(2.0))) ? (hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(t_1, hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (hypot(t_2, t_0) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((fma((dX_46_v ^ Float32(2.0)), (floor(h) ^ Float32(2.0)), (t_1 ^ Float32(2.0))) != fma((dX_46_v ^ Float32(2.0)), (floor(h) ^ Float32(2.0)), (t_1 ^ Float32(2.0)))) ? (hypot(t_2, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) : (((hypot(t_2, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) != (hypot(t_2, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0))) ? fma((dX_46_v ^ Float32(2.0)), (floor(h) ^ Float32(2.0)), (t_1 ^ Float32(2.0))) : max(fma((dX_46_v ^ Float32(2.0)), (floor(h) ^ Float32(2.0)), (t_1 ^ Float32(2.0))), (hypot(t_2, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 10

   1. Initial program 73.2%

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

   3. Taylor expanded in w around 0 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in dY.u around 0 67.5%

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

      1. *-commutative 40.1%

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

   8. Simplified 67.5%

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

   if 10 < dY.u

   1. Initial program 60.5%

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

   3. Taylor expanded in w around 0 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in dX.u around 0 58.3%

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

      1. fma-define 58.3%

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

      2. unpow2 58.3%

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

      3. unpow2 58.3%

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

      4. swap-sqr 58.3%

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

      5. unpow2 58.3%

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

   8. Simplified 58.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 10:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.v}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}\right), {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dY.w\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ \mathbf{if}\;dY.u \leq 4000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)) (t_1 (* (floor h) dY.v)))
   (if (<= dY.u 4000.0)
     (log2
      (sqrt
       (fmax
        (pow
         (hypot
          (* (floor d) dX.w)
          (hypot (* (floor w) dX.u) (* (floor h) dX.v)))
         2.0)
        (pow (hypot t_0 t_1) 2.0))))
     (log2
      (sqrt
       (fmax
        (* (pow (floor h) 2.0) (pow dX.v 2.0))
        (pow (hypot t_0 (hypot (* (floor w) dY.u) t_1)) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dY_46_w;
	float t_1 = floorf(h) * dY_46_v;
	float tmp;
	if (dY_46_u <= 4000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(hypotf(t_0, t_1), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(h), 2.0f) * powf(dX_46_v, 2.0f)), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), t_1)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w)
	t_1 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(4000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (hypot(t_0, t_1) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0)))) ? (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) : (((hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))) : max(Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	t_1 = floor(h) * dY_46_v;
	tmp = single(0.0);
	if (dY_46_u <= single(4000.0))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (hypot(t_0, t_1) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) ^ single(2.0)) * (dX_46_v ^ single(2.0))), (hypot(t_0, hypot((floor(w) * dY_46_u), t_1)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 4e3

   1. Initial program 73.5%

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

   3. Taylor expanded in w around 0 73.5%

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

   5. Simplified 73.5%

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

   6. Taylor expanded in dY.u around 0 68.0%

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

      1. *-commutative 41.1%

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

   8. Simplified 68.0%

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

   if 4e3 < dY.u

   1. Initial program 58.1%

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

   3. Taylor expanded in w around 0 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in dX.v around inf 53.1%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 4000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;dX.v \leq 30000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log t\_0}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_0, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u)))
   (if (<= dX.v 30000000.0)
     (log2
      (sqrt
       (fmax
        (exp (* 2.0 (log t_0)))
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot t_0 (* (floor h) dX.v)) 2.0)
        (* (pow (floor h) 2.0) (pow dY.v 2.0))))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float tmp;
	if (dX_46_v <= 30000000.0f) {
		tmp = log2f(sqrtf(fmaxf(expf((2.0f * logf(t_0))), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, (floorf(h) * dX_46_v)), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(30000000.0))
		tmp = log2(sqrt(((exp(Float32(Float32(2.0) * log(t_0))) != exp(Float32(Float32(2.0) * log(t_0)))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? exp(Float32(Float32(2.0) * log(t_0))) : max(exp(Float32(Float32(2.0) * log(t_0))), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(t_0, Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(30000000.0))
		tmp = log2(sqrt(max(exp((single(2.0) * log(t_0))), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, (floor(h) * dX_46_v)) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 3e7

   1. Initial program 68.8%

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

   3. Taylor expanded in w around 0 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dX.u around inf 58.4%

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

      1. pow-prod-down 58.4%

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

      2. pow-to-exp 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{e^{\log \left(dX.u \cdot \left\lfloorw\right\rfloor\right) \cdot 2}}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \]

      3. *-commutative 54.7%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{\log \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)} \cdot 2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \]

   8. Applied egg-rr 54.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{e^{\log \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot 2}}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \]

   if 3e7 < dX.v

   1. Initial program 79.8%

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

   3. Taylor expanded in w around 0 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in dY.v around inf 78.5%

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

      1. *-commutative 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in dX.w around 0 77.6%

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

       1. *-commutative 77.6%

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

       2. unpow2 77.6%

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

       3. unpow2 77.6%

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

       4. swap-sqr 77.6%

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

       5. unpow2 77.6%

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

       6. unpow2 77.6%

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

       7. swap-sqr 77.6%

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

       8. hypot-undefine 77.6%

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

   11. Simplified 77.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 30000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 30000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 30000000.0)
   (log2
    (sqrt
     (fmax
      (* (pow dX.u 2.0) (pow (floor w) 2.0))
      (pow
       (hypot (* (floor d) dY.w) (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
       2.0))))
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 30000000.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f)), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(30000000.0))
		tmp = log2(sqrt(((Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) != Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) : max(Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(30000000.0))
		tmp = log2(sqrt(max(((dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0))), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 3e7

   1. Initial program 68.8%

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

   3. Taylor expanded in w around 0 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dX.u around inf 58.4%

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

   if 3e7 < dX.v

   1. Initial program 79.8%

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

   3. Taylor expanded in w around 0 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in dY.v around inf 78.5%

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

      1. *-commutative 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in dX.w around 0 77.6%

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

       1. *-commutative 77.6%

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

       2. unpow2 77.6%

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

       3. unpow2 77.6%

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

       4. swap-sqr 77.6%

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

       5. unpow2 77.6%

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

       6. unpow2 77.6%

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

       7. swap-sqr 77.6%

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

       8. hypot-undefine 77.6%

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

   11. Simplified 77.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 30000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.u \leq 25000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {t\_0}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)))
   (if (<= dX.u 25000000.0)
     (log2
      (sqrt
       (fmax
        (* (pow (floor h) 2.0) (pow dX.v 2.0))
        (pow (hypot t_0 (hypot (* (floor w) dY.u) (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow
         (hypot
          (* (floor d) dX.w)
          (hypot (* (floor w) dX.u) (* (floor h) dX.v)))
         2.0)
        (pow t_0 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_u <= 25000000.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(h), 2.0f) * powf(dX_46_v, 2.0f)), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), powf(t_0, 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(25000000.0))
		tmp = log2(sqrt(((Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0)))) ? (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))) : max(Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (t_0 ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_u <= single(25000000.0))
		tmp = log2(sqrt(max(((floor(h) ^ single(2.0)) * (dX_46_v ^ single(2.0))), (hypot(t_0, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (t_0 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 2.5e7

   1. Initial program 71.2%

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

   3. Taylor expanded in w around 0 71.2%

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

   5. Simplified 71.2%

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

   6. Taylor expanded in dX.v around inf 62.5%

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

   if 2.5e7 < dX.u

   1. Initial program 64.7%

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

   3. Taylor expanded in w around 0 64.7%

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

   5. Simplified 64.7%

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

   6. Taylor expanded in dY.w around inf 61.8%

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

      1. *-commutative 28.0%

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

      2. unpow2 28.0%

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

      3. unpow2 28.0%

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

      4. swap-sqr 28.0%

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

      5. unpow2 28.0%

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

   8. Simplified 61.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 25000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ \mathbf{if}\;dY.u \leq 4000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, e^{2 \cdot \log t\_0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, t\_0\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v)))
   (if (<= dY.u 4000.0)
     (log2
      (sqrt
       (fmax
        (pow
         (hypot
          (* (floor d) dX.w)
          (hypot (* (floor w) dX.u) (* (floor h) dX.v)))
         2.0)
        (exp (* 2.0 (log t_0))))))
     (log2
      (sqrt
       (fmax
        (* (pow (floor h) 2.0) (pow dX.v 2.0))
        (pow
         (hypot (* (floor d) dY.w) (hypot (* (floor w) dY.u) t_0))
         2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float tmp;
	if (dY_46_u <= 4000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), expf((2.0f * logf(t_0))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(floorf(h), 2.0f) * powf(dX_46_v, 2.0f)), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), t_0)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(4000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? exp(Float32(Float32(2.0) * log(t_0))) : ((exp(Float32(Float32(2.0) * log(t_0))) != exp(Float32(Float32(2.0) * log(t_0)))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), exp(Float32(Float32(2.0) * log(t_0))))))));
	else
		tmp = log2(sqrt(((Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))) : max(Float32((floor(h) ^ Float32(2.0)) * (dX_46_v ^ Float32(2.0))), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dY_46_v;
	tmp = single(0.0);
	if (dY_46_u <= single(4000.0))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), exp((single(2.0) * log(t_0))))));
	else
		tmp = log2(sqrt(max(((floor(h) ^ single(2.0)) * (dX_46_v ^ single(2.0))), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), t_0)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 4e3

   1. Initial program 73.5%

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

   3. Taylor expanded in w around 0 73.5%

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

   5. Simplified 73.5%

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

   6. Taylor expanded in dY.v around inf 60.6%

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

      1. *-commutative 60.6%

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

   8. Simplified 60.6%

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

      1. pow-prod-down 60.6%

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

      2. pow-to-exp 55.2%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)\right)}^{2}, \color{blue}{e^{\log \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot 2}}\right)}\right) \]

   10. Applied egg-rr 55.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(dX.w \cdot \left\lfloord\right\rfloor, \mathsf{hypot}\left(dX.u \cdot \left\lfloorw\right\rfloor, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)\right)}^{2}, \color{blue}{e^{\log \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot 2}}\right)}\right) \]

   if 4e3 < dY.u

   1. Initial program 58.1%

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

   3. Taylor expanded in w around 0 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in dX.v around inf 53.1%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 4000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, e^{2 \cdot \log \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloorh\right\rfloor\right)}^{2}\\ \mathbf{if}\;dY.u \leq 4000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, t\_0 \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (floor h) 2.0)))
   (if (<= dY.u 4000.0)
     (log2
      (sqrt
       (fmax
        (pow
         (hypot
          (* (floor d) dX.w)
          (hypot (* (floor w) dX.u) (* (floor h) dX.v)))
         2.0)
        (* t_0 (pow dY.v 2.0)))))
     (log2
      (sqrt
       (fmax
        (* t_0 (pow dX.v 2.0))
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(floorf(h), 2.0f);
	float tmp;
	if (dY_46_u <= 4000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v))), 2.0f), (t_0 * powf(dY_46_v, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 * powf(dX_46_v, 2.0f)), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(4000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? Float32(t_0 * (dY_46_v ^ Float32(2.0))) : ((Float32(t_0 * (dY_46_v ^ Float32(2.0))) != Float32(t_0 * (dY_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), Float32(t_0 * (dY_46_v ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32(t_0 * (dX_46_v ^ Float32(2.0))) != Float32(t_0 * (dX_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? Float32(t_0 * (dX_46_v ^ Float32(2.0))) : max(Float32(t_0 * (dX_46_v ^ Float32(2.0))), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) ^ single(2.0);
	tmp = single(0.0);
	if (dY_46_u <= single(4000.0))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v))) ^ single(2.0)), (t_0 * (dY_46_v ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((t_0 * (dX_46_v ^ single(2.0))), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 4e3

   1. Initial program 73.5%

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

   3. Taylor expanded in w around 0 73.5%

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

   5. Simplified 73.5%

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

   6. Taylor expanded in dY.v around inf 60.6%

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

      1. *-commutative 60.6%

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

   8. Simplified 60.6%

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

   if 4e3 < dY.u

   1. Initial program 58.1%

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

   3. Taylor expanded in w around 0 58.1%

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

   5. Simplified 58.1%

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

   6. Taylor expanded in dX.v around inf 53.1%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 4000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dX.v}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 30000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 30000000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow
       (hypot (* (floor d) dY.w) (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
       2.0))))
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 30000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(30000000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(30000000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 3e7

   1. Initial program 68.8%

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

   3. Taylor expanded in w around 0 68.8%

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

   5. Simplified 68.8%

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

   6. Taylor expanded in dX.w around inf 55.8%

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

      1. unpow2 55.8%

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

      2. unpow2 55.8%

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

      3. swap-sqr 55.8%

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

      4. unpow2 55.8%

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

   8. Simplified 55.8%

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

   if 3e7 < dX.v

   1. Initial program 79.8%

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

   3. Taylor expanded in w around 0 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in dY.v around inf 78.5%

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

      1. *-commutative 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in dX.w around 0 77.6%

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

       1. *-commutative 77.6%

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

       2. unpow2 77.6%

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

       3. unpow2 77.6%

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

       4. swap-sqr 77.6%

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

       5. unpow2 77.6%

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

       6. unpow2 77.6%

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

       7. swap-sqr 77.6%

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

       8. hypot-undefine 77.6%

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

   11. Simplified 77.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 30000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 12000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 12000000.0)
   (log2
    (sqrt
     (fmax
      (* (pow dX.u 2.0) (pow (floor w) 2.0))
      (pow (hypot (* (floor w) dY.u) (* (floor h) dY.v)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 12000000.0f) {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f)), powf(hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(12000000.0))
		tmp = log2(sqrt(((Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) != Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) : max(Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))), (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(12000000.0))
		tmp = log2(sqrt(max(((dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0))), (hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.2e7

   1. Initial program 68.7%

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

   3. Taylor expanded in w around 0 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in dX.u around inf 58.6%

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

   7. Taylor expanded in dY.w around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

      4. swap-sqr 49.3%

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

      5. *-commutative 49.3%

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

      6. unpow2 49.3%

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

      7. unpow2 49.3%

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

      8. swap-sqr 49.3%

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

      9. rem-square-sqrt 49.3%

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

      10. hypot-undefine 49.3%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)} \cdot \sqrt{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right) \]

      11. hypot-undefine 49.3%

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

      12. unpow2 49.3%

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

   9. Simplified 49.3%

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

   if 1.2e7 < dX.v

   1. Initial program 80.5%

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

   3. Taylor expanded in w around 0 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in dY.v around inf 79.3%

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

      1. *-commutative 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in dX.v around inf 76.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 12000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.u \leq 100000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.u 100000.0)
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor d) dX.w) (* (floor h) dX.v)) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))
   (log2
    (sqrt
     (fmax
      (* (pow dX.u 2.0) (pow (floor w) 2.0))
      (pow (hypot (* (floor w) dY.u) (* (floor h) dY.v)) 2.0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dY_46_u <= 100000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), (floorf(h) * dX_46_v)), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f)), powf(hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(100000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(d) * dX_46_w), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) != Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) : max(Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))), (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dY_46_u <= single(100000.0))
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), (floor(h) * dX_46_v)) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(((dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0))), (hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 1e5

   1. Initial program 73.9%

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

   3. Taylor expanded in w around 0 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in dY.v around inf 60.5%

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

      1. *-commutative 60.5%

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

   8. Simplified 60.5%

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

   9. Taylor expanded in dX.u around 0 50.4%

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

       1. unpow2 50.4%

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

       2. unpow2 50.4%

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

       3. swap-sqr 50.4%

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

       4. unpow2 50.4%

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

       5. +-commutative 50.4%

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

       6. exp-to-pow 36.6%

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

       7. exp-lft-sqr 36.6%

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

       8. unpow2 36.6%

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

       9. unpow2 36.6%

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

       10. swap-sqr 36.6%

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

       11. hypot-define 36.6%

           \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left(\mathsf{hypot}\left(e^{\log \left(dX.w \cdot \left\lfloord\right\rfloor\right)}, dX.v \cdot \left\lfloorh\right\rfloor\right)\right)}}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right) \]

       12. rem-exp-log 50.4%

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

   11. Simplified 50.4%

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

   if 1e5 < dY.u

   1. Initial program 55.0%

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

   3. Taylor expanded in w around 0 55.0%

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

   5. Simplified 55.0%

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

   6. Taylor expanded in dX.u around inf 50.4%

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

   7. Taylor expanded in dY.w around 0 44.2%

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

      1. *-commutative 44.2%

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

      2. unpow2 44.2%

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

      3. unpow2 44.2%

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

      4. swap-sqr 44.2%

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

      5. *-commutative 44.2%

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

      6. unpow2 44.2%

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

      7. unpow2 44.2%

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

      8. swap-sqr 44.2%

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

      9. rem-square-sqrt 44.2%

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

      10. hypot-undefine 44.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)} \cdot \sqrt{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right) \]

      11. hypot-undefine 44.2%

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

      12. unpow2 44.2%

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

   9. Simplified 44.2%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 100000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.u \leq 2000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.u 2000.0)
   (log2
    (sqrt
     (fmax
      (pow (hypot (* (floor w) dX.u) (* (floor h) dX.v)) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow (hypot (* (floor d) dY.w) (* (floor w) dY.u)) 2.0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dY_46_u <= 2000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(h) * dX_46_v)), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (floorf(w) * dY_46_u)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(2000.0))
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(h) * dX_46_v)) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dY_46_u <= single(2000.0))
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(d) * dY_46_w), (floor(w) * dY_46_u)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2e3

   1. Initial program 73.4%

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

   3. Taylor expanded in w around 0 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in dY.v around inf 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   9. Taylor expanded in dX.w around 0 54.0%

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

       1. *-commutative 54.0%

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

       2. unpow2 54.0%

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

       3. unpow2 54.0%

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

       4. swap-sqr 54.0%

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

       5. unpow2 54.0%

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

       6. unpow2 54.0%

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

       7. swap-sqr 54.0%

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

       8. hypot-undefine 54.0%

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

   11. Simplified 54.0%

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

   if 2e3 < dY.u

   1. Initial program 58.8%

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

   3. Taylor expanded in w around 0 58.8%

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

   5. Simplified 58.8%

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

   6. Taylor expanded in dX.w around inf 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

      3. swap-sqr 51.4%

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

      4. unpow2 51.4%

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

   8. Simplified 51.4%

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

   9. Taylor expanded in dY.u around inf 49.8%

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

       1. *-commutative 42.5%

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

   11. Simplified 49.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 2000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 12000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 12000000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor d) dX.w) 2.0)
      (pow (hypot (* (floor d) dY.w) (* (floor w) dY.u)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 12000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (floorf(w) * dY_46_u)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(12000000.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), Float32(floor(w) * dY_46_u)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(12000000.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot((floor(d) * dY_46_w), (floor(w) * dY_46_u)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.2e7

   1. Initial program 68.7%

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

   3. Taylor expanded in w around 0 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in dX.w around inf 56.0%

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

      1. unpow2 56.0%

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

      2. unpow2 56.0%

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

      3. swap-sqr 56.0%

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

      4. unpow2 56.0%

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

   8. Simplified 56.0%

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

   9. Taylor expanded in dY.u around inf 46.7%

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

       1. *-commutative 38.2%

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

   11. Simplified 46.7%

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

   if 1.2e7 < dX.v

   1. Initial program 80.5%

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

   3. Taylor expanded in w around 0 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in dY.v around inf 79.3%

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

      1. *-commutative 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in dX.v around inf 76.7%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 12000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dY.u \leq 500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)))
   (if (<= dY.u 500.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor w) dX.u) 2.0)
        (pow (hypot t_0 (* (floor h) dY.v)) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (* (floor d) dX.w) 2.0)
        (pow (hypot t_0 (* (floor w) dY.u)) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dY_46_w;
	float tmp;
	if (dY_46_u <= 500.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), powf(hypotf(t_0, (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), powf(hypotf(t_0, (floorf(w) * dY_46_u)), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(500.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), (hypot(t_0, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), (hypot(t_0, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dY_46_u <= single(500.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (hypot(t_0, (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), (hypot(t_0, (floor(w) * dY_46_u)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 500

   1. Initial program 73.7%

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

   3. Taylor expanded in w around 0 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in dY.u around 0 68.1%

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

      1. *-commutative 41.4%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in dX.u around inf 51.2%

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

       1. *-commutative 51.2%

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

   11. Simplified 51.2%

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

   if 500 < dY.u

   1. Initial program 57.9%

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

   3. Taylor expanded in w around 0 57.9%

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

   5. Simplified 57.9%

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

   6. Taylor expanded in dX.w around inf 50.6%

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

      1. unpow2 50.6%

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

      2. unpow2 50.6%

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

      3. swap-sqr 50.6%

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

      4. unpow2 50.6%

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

   8. Simplified 50.6%

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

   9. Taylor expanded in dY.u around inf 49.1%

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

       1. *-commutative 42.4%

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

   11. Simplified 49.1%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 500:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dX.v \leq 12000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dX.v 12000000.0)
   (log2
    (sqrt
     (fmax
      (pow (* (floor w) dX.u) 2.0)
      (pow (hypot (* (floor w) dY.u) (* (floor h) dY.v)) 2.0))))
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dX_46_v <= 12000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), powf(hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(12000000.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), (hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dX_46_v <= single(12000000.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 1.2e7

   1. Initial program 68.7%

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

   3. Taylor expanded in w around 0 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in dX.u around inf 58.6%

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

   7. Taylor expanded in dY.w around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

      4. swap-sqr 49.3%

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

      5. *-commutative 49.3%

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

      6. unpow2 49.3%

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

      7. unpow2 49.3%

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

      8. swap-sqr 49.3%

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

      9. rem-square-sqrt 49.3%

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

      10. hypot-undefine 49.3%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)} \cdot \sqrt{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right) \]

      11. hypot-undefine 49.3%

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

      12. unpow2 49.3%

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

   9. Simplified 49.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{{\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. pow1 49.3%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\right)}^{1}}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right) \]

       2. pow-prod-down 49.3%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\color{blue}{\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}\right)}}^{1}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right) \]

       3. *-commutative 49.3%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left({\color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}}^{2}\right)}^{1}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right) \]

   11. Applied egg-rr 49.3%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\right)}^{1}}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right) \]
   12. Step-by-step derivation

       1. unpow1 49.3%

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

   13. Simplified 49.3%

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

   if 1.2e7 < dX.v

   1. Initial program 80.5%

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

   3. Taylor expanded in w around 0 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in dY.v around inf 79.3%

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

      1. *-commutative 79.3%

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

   8. Simplified 79.3%

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

   9. Taylor expanded in dX.v around inf 76.7%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 12000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.u \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.u 200.0)
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0)))
     (log2 (sqrt (fmax (exp (* 2.0 (log (* (floor w) 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 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_u <= 200.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(expf((2.0f * logf((floorf(w) * 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 = Float32(floor(d) * dY_46_w) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(200.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt(((exp(Float32(Float32(2.0) * log(Float32(floor(w) * dX_46_u)))) != exp(Float32(Float32(2.0) * log(Float32(floor(w) * dX_46_u))))) ? t_0 : ((t_0 != t_0) ? exp(Float32(Float32(2.0) * log(Float32(floor(w) * dX_46_u)))) : max(exp(Float32(Float32(2.0) * log(Float32(floor(w) * dX_46_u)))), 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) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(200.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(exp((single(2.0) * log((floor(w) * dX_46_u)))), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 200

   1. Initial program 71.6%

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

   3. Taylor expanded in w around 0 71.6%

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

   5. Simplified 71.6%

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

   6. Taylor expanded in dX.w around inf 57.9%

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

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. swap-sqr 57.9%

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

      4. unpow2 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in dY.w around inf 35.5%

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

       1. *-commutative 35.5%

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

       2. unpow2 35.5%

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

       3. unpow2 35.5%

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

       4. swap-sqr 35.5%

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

       5. unpow2 35.5%

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

   11. Simplified 35.5%

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

   if 200 < dX.u

   1. Initial program 64.5%

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

   3. Taylor expanded in w around 0 64.5%

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

   5. Simplified 64.5%

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

   6. Taylor expanded in dX.u around inf 54.2%

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

      1. pow-prod-down 54.2%

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

      2. pow-to-exp 53.9%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{e^{\log \left(dX.u \cdot \left\lfloorw\right\rfloor\right) \cdot 2}}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \]

      3. *-commutative 53.9%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(e^{\log \color{blue}{\left(\left\lfloorw\right\rfloor \cdot dX.u\right)} \cdot 2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \]

   8. Applied egg-rr 53.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\color{blue}{e^{\log \left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot 2}}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right) \]

   9. Taylor expanded in dY.w around inf 46.4%

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

       1. *-commutative 26.5%

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

       2. unpow2 26.5%

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

       3. unpow2 26.5%

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

       4. swap-sqr 26.5%

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

       5. unpow2 26.5%

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

   11. Simplified 46.4%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(e^{2 \cdot \log \left(\left\lfloorw\right\rfloor \cdot dX.u\right)}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\\ \mathbf{if}\;dX.u \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor d) dY.w) 2.0)))
   (if (<= dX.u 200.0)
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0)))
     (log2 (sqrt (fmax (* (pow dX.u 2.0) (pow (floor w) 2.0)) 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 = powf((floorf(d) * dY_46_w), 2.0f);
	float tmp;
	if (dX_46_u <= 200.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f)), 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) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(200.0))
		tmp = log2(sqrt((((Float32(floor(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt(((Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) != Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) ? t_0 : ((t_0 != t_0) ? Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) : max(Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))), 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) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(200.0))
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0))), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 200

   1. Initial program 71.6%

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

   3. Taylor expanded in w around 0 71.6%

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

   5. Simplified 71.6%

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

   6. Taylor expanded in dX.w around inf 57.9%

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

      1. unpow2 57.9%

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

      2. unpow2 57.9%

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

      3. swap-sqr 57.9%

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

      4. unpow2 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in dY.w around inf 35.5%

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

       1. *-commutative 35.5%

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

       2. unpow2 35.5%

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

       3. unpow2 35.5%

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

       4. swap-sqr 35.5%

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

       5. unpow2 35.5%

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

   11. Simplified 35.5%

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

   if 200 < dX.u

   1. Initial program 64.5%

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

   3. Taylor expanded in w around 0 64.5%

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

   5. Simplified 64.5%

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

   6. Taylor expanded in dX.u around inf 54.2%

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

   7. Taylor expanded in dY.w around inf 46.6%

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

      1. *-commutative 26.5%

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

      2. unpow2 26.5%

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

      3. unpow2 26.5%

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

      4. swap-sqr 26.5%

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

      5. unpow2 26.5%

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

   9. Simplified 46.6%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 200:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\\ \mathbf{if}\;dY.w \leq 300:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (pow dX.u 2.0) (pow (floor w) 2.0))))
   (if (<= dY.w 300.0)
     (log2 (sqrt (fmax t_0 (pow (* (floor h) dY.v) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (* (floor d) dY.w) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_w <= 300.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(h) * dY_46_v), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(d) * dY_46_w), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(300.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(h) * dY_46_v) ^ Float32(2.0)) : (((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0));
	tmp = single(0.0);
	if (dY_46_w <= single(300.0))
		tmp = log2(sqrt(max(t_0, ((floor(h) * dY_46_v) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(d) * dY_46_w) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 300

   1. Initial program 70.8%

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

   3. Taylor expanded in w around 0 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in dX.u around inf 55.5%

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

   7. Taylor expanded in dY.w around 0 50.1%

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

      1. *-commutative 50.1%

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

      2. unpow2 50.1%

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

      3. unpow2 50.1%

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

      4. swap-sqr 50.1%

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

      5. *-commutative 50.1%

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

      6. unpow2 50.1%

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

      7. unpow2 50.1%

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

      8. swap-sqr 50.1%

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

      9. rem-square-sqrt 50.1%

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

      10. hypot-undefine 50.1%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)} \cdot \sqrt{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right) \]

      11. hypot-undefine 50.1%

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

      12. unpow2 50.1%

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

   9. Simplified 50.1%

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

   10. Taylor expanded in dY.u around 0 39.2%

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

       1. *-commutative 39.2%

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

   12. Simplified 39.2%

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

   if 300 < dY.w

   1. Initial program 67.0%

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

   3. Taylor expanded in w around 0 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in dX.u around inf 59.2%

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

   7. Taylor expanded in dY.w around inf 50.0%

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

      1. *-commutative 45.0%

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

      2. unpow2 45.0%

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

      3. unpow2 45.0%

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

      4. swap-sqr 45.0%

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

      5. unpow2 45.0%

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

   9. Simplified 50.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 300:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 39.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}\\ \mathbf{if}\;dY.u \leq 2000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (pow dX.u 2.0) (pow (floor w) 2.0))))
   (if (<= dY.u 2000.0)
     (log2 (sqrt (fmax t_0 (pow (* (floor h) dY.v) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (* (floor w) dY.u) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f);
	float tmp;
	if (dY_46_u <= 2000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(h) * dY_46_v), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(w) * dY_46_u), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(2000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(h) * dY_46_v) ^ Float32(2.0)) : (((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(w) * dY_46_u) ^ Float32(2.0)) : (((Float32(floor(w) * dY_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dY_46_u) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(w) * dY_46_u) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0));
	tmp = single(0.0);
	if (dY_46_u <= single(2000.0))
		tmp = log2(sqrt(max(t_0, ((floor(h) * dY_46_v) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(w) * dY_46_u) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2e3

   1. Initial program 73.4%

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

   3. Taylor expanded in w around 0 73.4%

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

   5. Simplified 73.4%

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

   6. Taylor expanded in dX.u around inf 58.1%

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

   7. Taylor expanded in dY.w around 0 49.2%

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

      1. *-commutative 49.2%

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

      2. unpow2 49.2%

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

      3. unpow2 49.2%

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

      4. swap-sqr 49.2%

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

      5. *-commutative 49.2%

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

      6. unpow2 49.2%

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

      7. unpow2 49.2%

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

      8. swap-sqr 49.2%

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

      9. rem-square-sqrt 49.2%

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

      10. hypot-undefine 49.2%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)} \cdot \sqrt{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right) \]

      11. hypot-undefine 49.2%

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

      12. unpow2 49.2%

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

   9. Simplified 49.2%

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

   10. Taylor expanded in dY.u around 0 41.3%

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

       1. *-commutative 41.3%

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

   12. Simplified 41.3%

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

   if 2e3 < dY.u

   1. Initial program 58.8%

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

   3. Taylor expanded in w around 0 58.8%

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

   5. Simplified 58.8%

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

   6. Taylor expanded in dX.u around inf 49.8%

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

   7. Taylor expanded in dY.w around 0 42.9%

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

      1. *-commutative 42.9%

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

      2. unpow2 42.9%

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

      3. unpow2 42.9%

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

      4. swap-sqr 42.9%

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

      5. *-commutative 42.9%

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

      6. unpow2 42.9%

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

      7. unpow2 42.9%

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

      8. swap-sqr 42.9%

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

      9. rem-square-sqrt 42.9%

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

      10. hypot-undefine 42.9%

          \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, \color{blue}{\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)} \cdot \sqrt{\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)}\right)}\right) \]

      11. hypot-undefine 42.9%

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

      12. unpow2 42.9%

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

   9. Simplified 42.9%

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

   10. Taylor expanded in dY.u around inf 42.5%

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

       1. *-commutative 42.5%

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

   12. Simplified 42.5%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 2000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;dY.w \leq 0.004999999888241291:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (if (<= dY.w 0.004999999888241291)
   (log2
    (sqrt
     (fmax
      (pow (* (floor h) dX.v) 2.0)
      (* (pow (floor h) 2.0) (pow dY.v 2.0)))))
   (log2
    (sqrt
     (fmax
      (* (pow dX.u 2.0) (pow (floor w) 2.0))
      (pow (* (floor d) dY.w) 2.0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float tmp;
	if (dY_46_w <= 0.004999999888241291f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), (powf(floorf(h), 2.0f) * powf(dY_46_v, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_u, 2.0f) * powf(floorf(w), 2.0f)), powf((floorf(d) * dY_46_w), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_w <= Float32(0.004999999888241291))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) : ((Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))) != Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0)))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), Float32((floor(h) ^ Float32(2.0)) * (dY_46_v ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) != Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0)))) ? (Float32(floor(d) * dY_46_w) ^ Float32(2.0)) : (((Float32(floor(d) * dY_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dY_46_w) ^ Float32(2.0))) ? Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))) : max(Float32((dX_46_u ^ Float32(2.0)) * (floor(w) ^ Float32(2.0))), (Float32(floor(d) * dY_46_w) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = single(0.0);
	if (dY_46_w <= single(0.004999999888241291))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), ((floor(h) ^ single(2.0)) * (dY_46_v ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(((dX_46_u ^ single(2.0)) * (floor(w) ^ single(2.0))), ((floor(d) * dY_46_w) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.w < 0.00499999989

   1. Initial program 70.8%

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

   3. Taylor expanded in w around 0 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in dY.v around inf 57.9%

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

      1. *-commutative 57.9%

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

   8. Simplified 57.9%

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

   9. Taylor expanded in dX.v around inf 40.4%

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

   if 0.00499999989 < dY.w

   1. Initial program 67.9%

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

   3. Taylor expanded in w around 0 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in dX.u around inf 59.1%

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

   7. Taylor expanded in dY.w around inf 46.5%

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

      1. *-commutative 42.0%

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

      2. unpow2 42.0%

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

      3. unpow2 42.0%

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

      4. swap-sqr 42.0%

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

      5. unpow2 42.0%

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

   9. Simplified 46.5%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.w \leq 0.004999999888241291:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\left\lfloorh\right\rfloor\right)}^{2} \cdot {dY.v}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.u}^{2} \cdot {\left(\left\lfloorw\right\rfloor\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 35.2% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor d) dX.w) 2.0) (pow (* (floor d) dY.w) 2.0)))))

C

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

Julia

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

MATLAB

function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), ((floor(d) * dY_46_w) ^ single(2.0)))));
end

Derivation

1. Initial program 70.0%

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

3. Taylor expanded in w around 0 70.0%

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

5. Simplified 70.0%

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

6. Taylor expanded in dX.w around inf 53.8%

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

   1. unpow2 53.8%

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

   2. unpow2 53.8%

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

   3. swap-sqr 53.8%

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

   4. unpow2 53.8%

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

8. Simplified 53.8%

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

9. Taylor expanded in dY.w around inf 33.5%

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

    1. *-commutative 33.5%

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

    2. unpow2 33.5%

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

    3. unpow2 33.5%

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

    4. swap-sqr 33.5%

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

    5. unpow2 33.5%

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

11. Simplified 33.5%

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

12. Final simplification 33.5%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloord\right\rfloor \cdot dX.w\right)}^{2}, {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(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)))))))