Isotropic LOD (LOD)

Percentage Accurate: 67.9% → 67.9%

Time: 33.0s

Alternatives: 12

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: 67.9% 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: 67.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloord\right\rfloor \cdot dY.w\\ t_3 := \left\lfloord\right\rfloor \cdot dX.w\\ t_4 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_5 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_6 := \sqrt[3]{\log_{2} \left(\sqrt{\mathsf{max}\left({t\_4}^{2}, {\left(\mathsf{hypot}\left(t\_2, t\_0\right)\right)}^{2}\right)}\right)}\\ \mathbf{if}\;\mathsf{max}\left(\left(t\_4 \cdot t\_4 + t\_5 \cdot t\_5\right) + t\_3 \cdot t\_3, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_2 \cdot t\_2\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_3, \mathsf{hypot}\left(t\_4, t\_5\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_2, \mathsf{hypot}\left(t\_0, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot {t\_6}^{2}\\ \end{array} \end{array} \]

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 d) dY.w))
        (t_3 (* (floor d) dX.w))
        (t_4 (* (floor w) dX.u))
        (t_5 (* (floor h) dX.v))
        (t_6
         (cbrt (log2 (sqrt (fmax (pow t_4 2.0) (pow (hypot t_2 t_0) 2.0)))))))
   (if (<=
        (fmax
         (+ (+ (* t_4 t_4) (* t_5 t_5)) (* t_3 t_3))
         (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_2 t_2)))
        INFINITY)
     (log2
      (sqrt
       (fmax
        (pow (hypot t_3 (hypot t_4 t_5)) 2.0)
        (pow (hypot t_2 (hypot t_0 t_1)) 2.0))))
     (* t_6 (pow t_6 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(d) * dY_46_w;
	float t_3 = floorf(d) * dX_46_w;
	float t_4 = floorf(w) * dX_46_u;
	float t_5 = floorf(h) * dX_46_v;
	float t_6 = cbrtf(log2f(sqrtf(fmaxf(powf(t_4, 2.0f), powf(hypotf(t_2, t_0), 2.0f)))));
	float tmp;
	if (fmaxf((((t_4 * t_4) + (t_5 * t_5)) + (t_3 * t_3)), (((t_0 * t_0) + (t_1 * t_1)) + (t_2 * t_2))) <= ((float) INFINITY)) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_3, hypotf(t_4, t_5)), 2.0f), powf(hypotf(t_2, hypotf(t_0, t_1)), 2.0f))));
	} else {
		tmp = t_6 * powf(t_6, 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(d) * dY_46_w)
	t_3 = Float32(floor(d) * dX_46_w)
	t_4 = Float32(floor(w) * dX_46_u)
	t_5 = Float32(floor(h) * dX_46_v)
	t_6 = cbrt(log2(sqrt((((t_4 ^ Float32(2.0)) != (t_4 ^ 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))) ? (t_4 ^ Float32(2.0)) : max((t_4 ^ Float32(2.0)), (hypot(t_2, t_0) ^ Float32(2.0))))))))
	tmp = Float32(0.0)
	if (((Float32(Float32(Float32(t_4 * t_4) + Float32(t_5 * t_5)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_4 * t_4) + Float32(t_5 * t_5)) + Float32(t_3 * t_3))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) : ((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2))) ? Float32(Float32(Float32(t_4 * t_4) + Float32(t_5 * t_5)) + Float32(t_3 * t_3)) : max(Float32(Float32(Float32(t_4 * t_4) + Float32(t_5 * t_5)) + Float32(t_3 * t_3)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2))))) <= Float32(Inf))
		tmp = log2(sqrt((((hypot(t_3, hypot(t_4, t_5)) ^ Float32(2.0)) != (hypot(t_3, hypot(t_4, t_5)) ^ Float32(2.0))) ? (hypot(t_2, hypot(t_0, t_1)) ^ Float32(2.0)) : (((hypot(t_2, hypot(t_0, t_1)) ^ Float32(2.0)) != (hypot(t_2, hypot(t_0, t_1)) ^ Float32(2.0))) ? (hypot(t_3, hypot(t_4, t_5)) ^ Float32(2.0)) : max((hypot(t_3, hypot(t_4, t_5)) ^ Float32(2.0)), (hypot(t_2, hypot(t_0, t_1)) ^ Float32(2.0)))))));
	else
		tmp = Float32(t_6 * (t_6 ^ Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0 72.6%

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

   5. Simplified 72.6%

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

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

   1. Initial program 72.6%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt[3]{\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)} \cdot \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) \cdot \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)} \]

   4. Applied egg-rr 28.1%

      \[\leadsto \log_{2} \color{blue}{\left({\left({\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

   5. Taylor expanded in dX.u around inf 25.9%

      \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]

   6. Taylor expanded in dY.u around inf 24.4%

      \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{dY.u \cdot \left\lfloorw\right\rfloor}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]
   7. Step-by-step derivation

      1. *-commutative 24.4%

         \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{\left\lfloorw\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]

   8. Simplified 24.4%

      \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{\left\lfloorw\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]
   9. Step-by-step derivation

      1. add-cube-cbrt 24.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

   10. Applied egg-rr 50.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)} \cdot {\left(\sqrt[3]{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dY.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dY.w\\ t_4 := \left\lfloord\right\rfloor \cdot dX.w\\ t_5 := \left\lfloorw\right\rfloor \cdot dX.u\\ \mathbf{if}\;\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_3 \cdot t\_3\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_4, \mathsf{hypot}\left(t\_5, t\_2\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_3, \mathsf{hypot}\left(t\_0, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\log_{2} \left(\sqrt{\mathsf{max}\left({t\_5}^{2}, {\left(\mathsf{hypot}\left(t\_3, t\_0\right)\right)}^{2}\right)}\right)}\right)}^{3}\\ \end{array} \end{array} \]

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)))
        INFINITY)
     (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))))
     (pow
      (cbrt (log2 (sqrt (fmax (pow t_5 2.0) (pow (hypot t_3 t_0) 2.0)))))
      3.0))))

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))) <= ((float) INFINITY)) {
		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 = powf(cbrtf(log2f(sqrtf(fmaxf(powf(t_5, 2.0f), powf(hypotf(t_3, t_0), 2.0f))))), 3.0f);
	}
	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(Inf))
		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 = cbrt(log2(sqrt((((t_5 ^ Float32(2.0)) != (t_5 ^ Float32(2.0))) ? (hypot(t_3, t_0) ^ Float32(2.0)) : (((hypot(t_3, t_0) ^ Float32(2.0)) != (hypot(t_3, t_0) ^ Float32(2.0))) ? (t_5 ^ Float32(2.0)) : max((t_5 ^ Float32(2.0)), (hypot(t_3, t_0) ^ Float32(2.0)))))))) ^ Float32(3.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   3. Taylor expanded in w around 0 72.6%

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

   5. Simplified 72.6%

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

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

   1. Initial program 72.6%

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

      1. add-cbrt-cube 28.4%

         \[\leadsto \log_{2} \color{blue}{\left(\sqrt[3]{\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)} \cdot \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) \cdot \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)} \]

   4. Applied egg-rr 28.1%

      \[\leadsto \log_{2} \color{blue}{\left({\left({\left(\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]

   5. Taylor expanded in dX.u around inf 25.9%

      \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\color{blue}{\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]

   6. Taylor expanded in dY.u around inf 24.4%

      \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{dY.u \cdot \left\lfloorw\right\rfloor}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]
   7. Step-by-step derivation

      1. *-commutative 24.4%

         \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{\left\lfloorw\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]

   8. Simplified 24.4%

      \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{\left\lfloorw\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]
   9. Step-by-step derivation

      1. add-cube-cbrt 24.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \cdot \sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}\right) \cdot \sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}} \]

      2. pow3 24.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)}\right)}^{3}} \]

   10. Applied egg-rr 50.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\left\lfloorw\right\rfloor \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right) \leq \infty:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloord\right\rfloor \cdot dY.w\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := \left\lfloord\right\rfloor \cdot dX.w\\ t_3 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_4 := \left\lfloorw\right\rfloor \cdot dY.u\\ \mathbf{if}\;dY.v \leq 99999997952:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_3 \cdot t\_3 + t\_1 \cdot t\_1\right) + t\_2 \cdot t\_2, {t\_4}^{2} + {t\_0}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_1}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(t\_4, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w))
        (t_1 (* (floor h) dX.v))
        (t_2 (* (floor d) dX.w))
        (t_3 (* (floor w) dX.u))
        (t_4 (* (floor w) dY.u)))
   (if (<= dY.v 99999997952.0)
     (log2
      (sqrt
       (fmax
        (+ (+ (* t_3 t_3) (* t_1 t_1)) (* t_2 t_2))
        (+ (pow t_4 2.0) (pow t_0 2.0)))))
     (log2
      (sqrt
       (fmax
        (pow t_1 2.0)
        (pow (hypot t_0 (hypot t_4 (* (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 = floorf(d) * dY_46_w;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = floorf(d) * dX_46_w;
	float t_3 = floorf(w) * dX_46_u;
	float t_4 = floorf(w) * dY_46_u;
	float tmp;
	if (dY_46_v <= 99999997952.0f) {
		tmp = log2f(sqrtf(fmaxf((((t_3 * t_3) + (t_1 * t_1)) + (t_2 * t_2)), (powf(t_4, 2.0f) + powf(t_0, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_1, 2.0f), powf(hypotf(t_0, hypotf(t_4, (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 = Float32(floor(d) * dY_46_w)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(d) * dX_46_w)
	t_3 = Float32(floor(w) * dX_46_u)
	t_4 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(99999997952.0))
		tmp = log2(sqrt(((Float32(Float32(Float32(t_3 * t_3) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) != Float32(Float32(Float32(t_3 * t_3) + Float32(t_1 * t_1)) + Float32(t_2 * t_2))) ? Float32((t_4 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) : ((Float32((t_4 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))) != Float32((t_4 ^ Float32(2.0)) + (t_0 ^ Float32(2.0)))) ? Float32(Float32(Float32(t_3 * t_3) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) : max(Float32(Float32(Float32(t_3 * t_3) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)), Float32((t_4 ^ Float32(2.0)) + (t_0 ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt((((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(t_0, hypot(t_4, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_0, hypot(t_4, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(t_4, Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : max((t_1 ^ Float32(2.0)), (hypot(t_0, hypot(t_4, 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(d) * dY_46_w;
	t_1 = floor(h) * dX_46_v;
	t_2 = floor(d) * dX_46_w;
	t_3 = floor(w) * dX_46_u;
	t_4 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if (dY_46_v <= single(99999997952.0))
		tmp = log2(sqrt(max((((t_3 * t_3) + (t_1 * t_1)) + (t_2 * t_2)), ((t_4 ^ single(2.0)) + (t_0 ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((t_1 ^ single(2.0)), (hypot(t_0, hypot(t_4, (floor(h) * dY_46_v))) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 99999998000

   1. Initial program 75.0%

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

      1. add-cbrt-cube 75.0%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. pow1/3 75.1%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. pow3 75.1%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. Applied egg-rr 75.1%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Taylor expanded in dY.v around 0 69.1%

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

      1. *-commutative 69.1%

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

      2. unpow2 69.1%

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

      3. unpow2 69.1%

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

      4. swap-sqr 69.1%

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

      5. unpow2 69.1%

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

      6. *-commutative 69.1%

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

      7. unpow2 69.1%

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

      8. unpow2 69.1%

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

      9. swap-sqr 69.1%

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

      10. unpow2 69.1%

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

   7. Simplified 69.1%

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

   if 99999998000 < dY.v

   1. Initial program 51.5%

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

   3. Taylor expanded in w around 0 51.5%

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

   5. Simplified 51.5%

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

   6. Taylor expanded in dX.v around inf 52.1%

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

      1. unpow2 52.1%

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

      2. unpow2 52.1%

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

      3. swap-sqr 52.1%

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

      4. unpow2 52.1%

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

   8. Simplified 52.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 99999997952:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2} + {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_2 := \left\lfloord\right\rfloor \cdot dX.w\\ t_3 := \left\lfloorw\right\rfloor \cdot dY.u\\ \mathbf{if}\;dX.v \leq 0.4000000059604645:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(t\_3, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right) + t\_2 \cdot t\_2, {t\_3}^{2}\right)}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* (floor h) dX.v))
        (t_2 (* (floor d) dX.w))
        (t_3 (* (floor w) dY.u)))
   (if (<= dX.v 0.4000000059604645)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot (* (floor d) dY.w) (hypot t_3 (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt
       (fmax (+ (+ (* t_0 t_0) (* t_1 t_1)) (* t_2 t_2)) (pow t_3 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(h) * dX_46_v;
	float t_2 = floorf(d) * dX_46_w;
	float t_3 = floorf(w) * dY_46_u;
	float tmp;
	if (dX_46_v <= 0.4000000059604645f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf(t_3, (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_1 * t_1)) + (t_2 * t_2)), powf(t_3, 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(d) * dX_46_w)
	t_3 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(0.4000000059604645))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(t_3, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(t_3, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(t_3, Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(t_3, Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2))) ? (t_3 ^ Float32(2.0)) : (((t_3 ^ Float32(2.0)) != (t_3 ^ Float32(2.0))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)) : max(Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_2 * t_2)), (t_3 ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(w) * dX_46_u;
	t_1 = floor(h) * dX_46_v;
	t_2 = floor(d) * dX_46_w;
	t_3 = floor(w) * dY_46_u;
	tmp = single(0.0);
	if (dX_46_v <= single(0.4000000059604645))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot(t_3, (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((((t_0 * t_0) + (t_1 * t_1)) + (t_2 * t_2)), (t_3 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.400000006

   1. Initial program 75.1%

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

   3. Taylor expanded in w around 0 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in dX.u around inf 62.8%

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

      1. unpow2 62.8%

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

      2. unpow2 62.8%

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

      3. swap-sqr 62.8%

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

      4. unpow2 62.8%

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

   8. Simplified 62.8%

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

   if 0.400000006 < dX.v

   1. Initial program 63.8%

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

      1. add-cbrt-cube 63.8%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. pow1/3 63.9%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. pow3 63.9%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. Applied egg-rr 63.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Taylor expanded in dY.u around inf 58.6%

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

      1. *-commutative 58.6%

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

      2. unpow2 58.6%

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

      3. unpow2 58.6%

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

      4. swap-sqr 58.6%

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

      5. unpow2 58.6%

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

   7. Simplified 58.6%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.4000000059604645:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), {\left(\left\lfloorw\right\rfloor \cdot dY.u\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloord\right\rfloor \cdot dY.w\\ t_2 := \left\lfloorh\right\rfloor \cdot dX.v\\ t_3 := \left\lfloord\right\rfloor \cdot dX.w\\ \mathbf{if}\;dX.v \leq 0.8199999928474426:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(\mathsf{hypot}\left(t\_1, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right) + t\_3 \cdot t\_3, {t\_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* (floor d) dY.w))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dX.w)))
   (if (<= dX.v 0.8199999928474426)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot t_1 (hypot (* (floor w) dY.u) (* (floor h) dY.v))) 2.0))))
     (log2
      (sqrt
       (fmax (+ (+ (* t_0 t_0) (* t_2 t_2)) (* t_3 t_3)) (pow 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(w) * dX_46_u;
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dX_46_w;
	float tmp;
	if (dX_46_v <= 0.8199999928474426f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_2 * t_2)) + (t_3 * t_3)), powf(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(w) * dX_46_u)
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(0.8199999928474426))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3)) != Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3)) : max(Float32(Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)) + Float32(t_3 * t_3)), (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(w) * dX_46_u;
	t_1 = floor(d) * dY_46_w;
	t_2 = floor(h) * dX_46_v;
	t_3 = floor(d) * dX_46_w;
	tmp = single(0.0);
	if (dX_46_v <= single(0.8199999928474426))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot(t_1, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((((t_0 * t_0) + (t_2 * t_2)) + (t_3 * t_3)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.819999993

   1. Initial program 75.2%

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

   3. Taylor expanded in w around 0 75.2%

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

   5. Simplified 75.2%

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

   6. Taylor expanded in dX.u around inf 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. swap-sqr 63.0%

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

      4. unpow2 63.0%

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

   8. Simplified 63.0%

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

   if 0.819999993 < dX.v

   1. Initial program 63.2%

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

      1. add-cbrt-cube 63.2%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. pow1/3 63.3%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. pow3 63.3%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. Applied egg-rr 63.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Taylor expanded in dY.w around inf 56.1%

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

      1. *-commutative 56.1%

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

      2. unpow2 56.1%

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

      3. unpow2 56.1%

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

      4. swap-sqr 56.1%

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

      5. unpow2 56.1%

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

   7. Simplified 56.1%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.8199999928474426:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), {\left(\left\lfloord\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ \mathbf{if}\;dX.v \leq 0.8199999928474426:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(t\_0, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {t\_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

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)) (t_1 (* (floor h) dY.v)))
   (if (<= dX.v 0.8199999928474426)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot (* (floor d) dY.w) (hypot (* (floor w) dY.u) t_1)) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor d) dX.w) (hypot t_0 (* (floor h) dX.v))) 2.0)
        (pow 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(w) * dX_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float tmp;
	if (dX_46_v <= 0.8199999928474426f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), t_1)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(d) * dX_46_w), hypotf(t_0, (floorf(h) * dX_46_v))), 2.0f), powf(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(w) * dX_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(0.8199999928474426))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)) : max((hypot(Float32(floor(d) * dX_46_w), hypot(t_0, Float32(floor(h) * dX_46_v))) ^ Float32(2.0)), (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(w) * dX_46_u;
	t_1 = floor(h) * dY_46_v;
	tmp = single(0.0);
	if (dX_46_v <= single(0.8199999928474426))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), t_1)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(d) * dX_46_w), hypot(t_0, (floor(h) * dX_46_v))) ^ single(2.0)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 0.819999993

   1. Initial program 75.2%

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

   3. Taylor expanded in w around 0 75.2%

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

   5. Simplified 75.2%

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

   6. Taylor expanded in dX.u around inf 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. swap-sqr 63.0%

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

      4. unpow2 63.0%

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

   8. Simplified 63.0%

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

   if 0.819999993 < dX.v

   1. Initial program 63.2%

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

      1. add-cbrt-cube 63.2%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. pow1/3 63.3%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. pow3 63.3%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. Applied egg-rr 63.3%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Taylor expanded in dY.v around inf 57.0%

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

      1. *-commutative 57.0%

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

      2. unpow2 57.0%

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

      3. unpow2 57.0%

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

      4. swap-sqr 57.0%

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

      5. unpow2 57.0%

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

   7. Simplified 57.0%

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

      1. add-sqr-sqrt 57.0%

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

   9. Applied egg-rr 57.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 0.8199999928474426:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dX.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\\ \mathbf{if}\;dX.u \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0
         (pow
          (hypot
           (* (floor d) dY.w)
           (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
          2.0)))
   (if (<= dX.u 0.5)
     (log2 (sqrt (fmax (pow (* (floor h) dX.v) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* (floor w) dX.u) 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(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f);
	float tmp;
	if (dX_46_u <= 0.5f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 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 = hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(0.5))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), t_0)))));
	else
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ 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 = hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(0.5))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 0.5

   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 73.8%

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

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

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

      1. unpow2 60.3%

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

      2. unpow2 60.3%

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

      3. swap-sqr 60.3%

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

      4. unpow2 60.3%

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

   8. Simplified 60.3%

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

   if 0.5 < dX.u

   1. Initial program 69.7%

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

   3. Taylor expanded in w around 0 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in dX.u around inf 62.7%

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

      1. unpow2 62.7%

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

      2. unpow2 62.7%

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

      3. swap-sqr 62.7%

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

      4. unpow2 62.7%

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

   8. Simplified 62.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 0.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorh\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorw\right\rfloor \cdot dX.u\\ t_1 := \left\lfloorh\right\rfloor \cdot dY.v\\ \mathbf{if}\;dX.v \leq 3600000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(t\_0, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {t\_1}^{2}\right)}\right)\\ \end{array} \end{array} \]

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)) (t_1 (* (floor h) dY.v)))
   (if (<= dX.v 3600000.0)
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow (hypot (* (floor d) dY.w) (hypot (* (floor w) dY.u) t_1)) 2.0))))
     (log2
      (sqrt (fmax (pow (hypot t_0 (* (floor h) dX.v)) 2.0) (pow 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(w) * dX_46_u;
	float t_1 = floorf(h) * dY_46_v;
	float tmp;
	if (dX_46_v <= 3600000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), t_1)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(t_0, (floorf(h) * dX_46_v)), 2.0f), powf(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(w) * dX_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(3600000.0))
		tmp = log2(sqrt((((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : max((t_0 ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), t_1)) ^ 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))) ? (t_1 ^ Float32(2.0)) : (((t_1 ^ Float32(2.0)) != (t_1 ^ 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)), (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(w) * dX_46_u;
	t_1 = floor(h) * dY_46_v;
	tmp = single(0.0);
	if (dX_46_v <= single(3600000.0))
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), t_1)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot(t_0, (floor(h) * dX_46_v)) ^ single(2.0)), (t_1 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 3.6e6

   1. Initial program 76.6%

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

   3. Taylor expanded in w around 0 76.6%

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

   5. Simplified 76.6%

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

   6. Taylor expanded in dX.u around inf 62.9%

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

      1. unpow2 62.9%

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

      2. unpow2 62.9%

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

      3. swap-sqr 62.9%

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

      4. unpow2 62.9%

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

   8. Simplified 62.9%

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

   if 3.6e6 < dX.v

   1. Initial program 44.5%

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

      1. add-cbrt-cube 44.5%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. pow1/3 44.5%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. pow3 44.5%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. Applied egg-rr 44.5%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Taylor expanded in dY.v around inf 41.8%

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

      1. *-commutative 41.8%

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

      2. unpow2 41.8%

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

      3. unpow2 41.8%

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

      4. swap-sqr 41.8%

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

      5. unpow2 41.8%

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

   7. Simplified 41.8%

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

   8. Taylor expanded in dX.w around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. unpow2 43.4%

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

      3. unpow2 43.4%

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

      4. swap-sqr 43.4%

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

      5. *-commutative 43.4%

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

      6. unpow2 43.4%

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

      7. unpow2 43.4%

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

      8. swap-sqr 43.4%

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

      9. unpow2 43.4%

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

      10. unpow2 43.4%

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

      11. rem-square-sqrt 43.4%

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

   10. Simplified 43.4%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 3600000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dY.u, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloorw\right\rfloor \cdot dX.u, \left\lfloorh\right\rfloor \cdot dX.v\right)\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}\\ t_1 := \left\lfloord\right\rfloor \cdot dY.w\\ \mathbf{if}\;dY.u \leq 0.05000000074505806:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\mathsf{hypot}\left(t\_1, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\mathsf{hypot}\left(t\_1, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dX.u) 2.0)) (t_1 (* (floor d) dY.w)))
   (if (<= dY.u 0.05000000074505806)
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (floor h) dY.v)) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (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((floorf(w) * dX_46_u), 2.0f);
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dY_46_u <= 0.05000000074505806f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (floorf(h) * dY_46_v)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (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(w) * dX_46_u) ^ Float32(2.0)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(0.05000000074505806))
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(t_1, Float32(floor(h) * dY_46_v)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) : (((hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0)) != (hypot(t_1, Float32(floor(w) * dY_46_u)) ^ Float32(2.0))) ? t_0 : max(t_0, (hypot(t_1, 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(w) * dX_46_u) ^ single(2.0);
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dY_46_u <= single(0.05000000074505806))
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(h) * dY_46_v)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(w) * dY_46_u)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 0.0500000007

   1. Initial program 74.7%

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

   3. Taylor expanded in w around 0 74.7%

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

   5. Simplified 74.7%

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

   6. Taylor expanded in dX.u around inf 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. swap-sqr 59.5%

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

      4. unpow2 59.5%

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

   8. Simplified 59.5%

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

   9. Taylor expanded in dY.u around 0 52.7%

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

       1. *-commutative 52.7%

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

   11. Simplified 52.7%

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

   if 0.0500000007 < dY.u

   1. Initial program 66.5%

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

   3. Taylor expanded in w around 0 66.5%

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

   5. Simplified 66.5%

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

   6. Taylor expanded in dX.u around inf 59.7%

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

      1. unpow2 59.7%

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

      2. unpow2 59.7%

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

      3. swap-sqr 59.7%

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

      4. unpow2 59.7%

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

   8. Simplified 59.7%

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

   9. Taylor expanded in dY.u around inf 56.9%

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

       1. *-commutative 24.0%

          \[\leadsto \log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.u \cdot \left\lfloorw\right\rfloor\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \color{blue}{\left\lfloorw\right\rfloor \cdot dY.u}\right)\right)}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right) \]

   11. Simplified 56.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 0.05000000074505806:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorw\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloorh\right\rfloor \cdot dY.v\\ \mathbf{if}\;dX.v \leq 200000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, t\_0\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {t\_0}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v)))
   (if (<= dX.v 200000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor w) dX.u) 2.0)
        (pow (hypot (* (floor d) dY.w) t_0) 2.0))))
     (log2
      (sqrt (fmax (* (pow dX.v 2.0) (pow (floor h) 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(h) * dY_46_v;
	float tmp;
	if (dX_46_v <= 200000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), powf(hypotf((floorf(d) * dY_46_w), t_0), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_v, 2.0f) * powf(floorf(h), 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(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(200000000.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), t_0) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), t_0) ^ 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(d) * dY_46_w), t_0) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0))) != Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0)))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ Float32(2.0))) : max(Float32((dX_46_v ^ Float32(2.0)) * (floor(h) ^ 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(h) * dY_46_v;
	tmp = single(0.0);
	if (dX_46_v <= single(200000000.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (hypot((floor(d) * dY_46_w), t_0) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((dX_46_v ^ single(2.0)) * (floor(h) ^ single(2.0))), (t_0 ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 2e8

   1. Initial program 75.6%

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

   3. Taylor expanded in w around 0 75.6%

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

   5. Simplified 75.7%

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

   6. Taylor expanded in dX.u around inf 62.2%

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

      1. unpow2 62.2%

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

      2. unpow2 62.2%

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

      3. swap-sqr 62.2%

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

      4. unpow2 62.2%

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

   8. Simplified 62.2%

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

   9. Taylor expanded in dY.u around 0 50.7%

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

       1. *-commutative 50.7%

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

   11. Simplified 50.7%

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

   if 2e8 < dX.v

   1. Initial program 48.4%

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

      1. add-cbrt-cube 48.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      2. pow1/3 48.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

      3. pow3 48.4%

         \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   4. Applied egg-rr 48.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   5. Taylor expanded in dY.v around inf 44.6%

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

      1. *-commutative 44.6%

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

      2. unpow2 44.6%

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

      3. unpow2 44.6%

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

      4. swap-sqr 44.6%

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

      5. unpow2 44.6%

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

   7. Simplified 44.6%

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

   8. Taylor expanded in dX.v around inf 41.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 200000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloorw\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloord\right\rfloor \cdot dY.w, \left\lfloorh\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.v}^{2} \cdot {\left(\left\lfloorh\right\rfloor\right)}^{2}, {\left(\left\lfloorh\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (* (pow dX.v 2.0) (pow (floor h) 2.0))
    (pow (* (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) {
	return log2f(sqrtf(fmaxf((powf(dX_46_v, 2.0f) * powf(floorf(h), 2.0f)), powf((floorf(h) * dY_46_v), 2.0f))));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 72.6%

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

   1. add-cbrt-cube 72.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. pow1/3 72.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. pow3 72.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

4. Applied egg-rr 72.6%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

5. Taylor expanded in dY.v around inf 54.5%

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

   1. *-commutative 54.5%

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

   2. unpow2 54.5%

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

   3. unpow2 54.5%

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

   4. swap-sqr 54.5%

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

   5. unpow2 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in dX.v around inf 36.3%

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

Alternative 12: 35.6% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor h) dX.v) 2.0) (pow (* (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) {
	return log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf((floorf(h) * dY_46_v), 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(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (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))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (Float32(floor(h) * dY_46_v) ^ 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(h) * dX_46_v) ^ single(2.0)), ((floor(h) * dY_46_v) ^ single(2.0)))));
end

Derivation

1. Initial program 72.6%

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

   1. add-cbrt-cube 72.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{\sqrt[3]{\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   2. pow1/3 72.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left(\left(\left\lfloorw\right\rfloor \cdot \left\lfloorw\right\rfloor\right) \cdot \left\lfloorw\right\rfloor\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

   3. pow3 72.6%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left({\color{blue}{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}}^{0.3333333333333333} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

4. Applied egg-rr 72.6%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\left(\left(\left\lfloorw\right\rfloor \cdot dX.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dX.u\right) + \left(\left\lfloorh\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dX.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dX.w\right), \left(\left(\color{blue}{{\left({\left(\left\lfloorw\right\rfloor\right)}^{3}\right)}^{0.3333333333333333}} \cdot dY.u\right) \cdot \left(\left\lfloorw\right\rfloor \cdot dY.u\right) + \left(\left\lfloorh\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloorh\right\rfloor \cdot dY.v\right)\right) + \left(\left\lfloord\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloord\right\rfloor \cdot dY.w\right)\right)}\right) \]

5. Taylor expanded in dY.v around inf 54.5%

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

   1. *-commutative 54.5%

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

   2. unpow2 54.5%

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

   3. unpow2 54.5%

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

   4. swap-sqr 54.5%

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

   5. unpow2 54.5%

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

7. Simplified 54.5%

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

8. Taylor expanded in dX.v around inf 36.3%

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

   1. *-commutative 36.3%

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

   2. unpow2 36.3%

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

   3. unpow2 36.3%

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

   4. swap-sqr 36.3%

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

   5. unpow2 36.3%

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

   6. *-commutative 36.3%

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

10. Simplified 36.3%

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

11. Final simplification 36.3%

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

Reproduce

herbie shell --seed 2024132 
(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)))))))