Isotropic LOD (LOD)

Percentage Accurate: 66.7% → 66.7%

Time: 46.1s

Alternatives: 17

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v)
	t_2 = Float32(floor(h) * dX_46_v)
	t_3 = Float32(floor(d) * dY_46_w)
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(w) * dX_46_u)
	return log2(sqrt(((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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 66.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor 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 (hypot t_5 t_2) t_4) 2.0)
        (pow (hypot t_3 (hypot t_0 t_1)) 2.0))))
     (log2 (sqrt (fmax (pow t_5 2.0) (pow (* (floor d) (- dY.w)) 2.0)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (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 64.9%

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

   3. Taylor expanded in w around 0 64.9%

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

   5. Simplified 64.9%

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

   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 64.9%

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

   3. Taylor expanded in w around 0 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in dY.u around 0 59.7%

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

      1. *-commutative 59.7%

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

   8. Simplified 59.7%

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

   9. Taylor expanded in dX.u around inf 42.7%

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

       1. unpow2 42.7%

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

       2. unpow2 42.7%

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

       3. swap-sqr 42.7%

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

       4. unpow2 42.7%

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

   11. Simplified 42.7%

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

   12. Taylor expanded in dY.w around -inf 33.6%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\color{blue}{\left(-1 \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right)\right)}}^{2}\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg 33.6%

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

       2. *-commutative 33.6%

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

       3. distribute-rgt-neg-in 33.6%

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

   14. Simplified 33.6%

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

4. Final simplification 64.9%

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

Alternative 2: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\\ \mathbf{if}\;dY.u \leq 200000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(t\_1, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_2\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left({dX.w}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2}, {t\_1}^{2}\right), {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_2\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

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 w) dX.u))
        (t_2 (* (floor h) dY.v)))
   (if (<= dY.u 200000.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot t_1 (* (floor h) dX.v)) (* (floor d) dX.w)) 2.0)
        (pow (hypot t_0 t_2) 2.0))))
     (log2
      (sqrt
       (fmax
        (fma (pow dX.w 2.0) (pow (floor d) 2.0) (pow t_1 2.0))
        (pow (hypot t_0 (hypot (* (floor w) dY.u) t_2)) 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(w) * dX_46_u;
	float t_2 = floorf(h) * dY_46_v;
	float tmp;
	if (dY_46_u <= 200000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf(t_1, (floorf(h) * dX_46_v)), (floorf(d) * dX_46_w)), 2.0f), powf(hypotf(t_0, t_2), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf(powf(dX_46_w, 2.0f), powf(floorf(d), 2.0f), powf(t_1, 2.0f)), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), t_2)), 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(w) * dX_46_u)
	t_2 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(200000.0))
		tmp = log2(sqrt((((hypot(hypot(t_1, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) != (hypot(hypot(t_1, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0))) ? (hypot(t_0, t_2) ^ Float32(2.0)) : (((hypot(t_0, t_2) ^ Float32(2.0)) != (hypot(t_0, t_2) ^ Float32(2.0))) ? (hypot(hypot(t_1, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) : max((hypot(hypot(t_1, Float32(floor(h) * dX_46_v)), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)), (hypot(t_0, t_2) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((fma((dX_46_w ^ Float32(2.0)), (floor(d) ^ Float32(2.0)), (t_1 ^ Float32(2.0))) != fma((dX_46_w ^ Float32(2.0)), (floor(d) ^ Float32(2.0)), (t_1 ^ Float32(2.0)))) ? (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_2)) ^ Float32(2.0)) : (((hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_2)) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_2)) ^ Float32(2.0))) ? fma((dX_46_w ^ Float32(2.0)), (floor(d) ^ Float32(2.0)), (t_1 ^ Float32(2.0))) : max(fma((dX_46_w ^ Float32(2.0)), (floor(d) ^ Float32(2.0)), (t_1 ^ Float32(2.0))), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_2)) ^ Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2e5

   1. Initial program 68.5%

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

   3. Taylor expanded in w around 0 68.5%

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

   5. Simplified 68.5%

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

   6. Taylor expanded in dY.u around 0 64.4%

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

      1. *-commutative 64.4%

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

   8. Simplified 64.4%

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

   if 2e5 < dY.u

   1. Initial program 48.0%

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

   3. Taylor expanded in w around 0 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in dX.v around 0 48.6%

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

      1. +-commutative 48.6%

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

      2. fma-define 48.6%

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

      3. unpow2 48.6%

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

      4. unpow2 48.6%

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

      5. swap-sqr 48.6%

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

      6. unpow2 48.6%

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

   8. Simplified 48.6%

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

4. Final simplification 61.6%

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

Alternative 3: 61.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_2 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.u \leq 200000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), t\_2\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_2}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w))
        (t_1 (* (floor h) dY.v))
        (t_2 (* (floor d) dX.w)))
   (if (<= dY.u 200000.0)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot (* (floor w) dX.u) (* (floor h) dX.v)) t_2) 2.0)
        (pow (hypot t_0 t_1) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow t_2 2.0)
        (pow (hypot t_0 (hypot (* (floor w) dY.u) t_1)) 2.0)))))))

C

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

Julia

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

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	t_1 = floor(h) * dY_46_v;
	t_2 = floor(d) * dX_46_w;
	tmp = single(0.0);
	if (dY_46_u <= single(200000.0))
		tmp = log2(sqrt(max((hypot(hypot((floor(w) * dX_46_u), (floor(h) * dX_46_v)), t_2) ^ single(2.0)), (hypot(t_0, t_1) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_2 ^ single(2.0)), (hypot(t_0, hypot((floor(w) * dY_46_u), t_1)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2e5

   1. Initial program 68.5%

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

   3. Taylor expanded in w around 0 68.5%

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

   5. Simplified 68.5%

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

   6. Taylor expanded in dY.u around 0 64.4%

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

      1. *-commutative 64.4%

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

   8. Simplified 64.4%

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

   if 2e5 < dY.u

   1. Initial program 48.0%

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

   3. Taylor expanded in w around 0 48.0%

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

   5. Simplified 48.0%

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

   6. Taylor expanded in dX.v around 0 48.6%

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

      1. +-commutative 48.6%

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

      2. fma-define 48.6%

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

      3. unpow2 48.6%

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

      4. unpow2 48.6%

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

      5. swap-sqr 48.6%

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

      6. unpow2 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in dX.w around inf 48.5%

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

       1. unpow2 48.5%

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

       2. unpow2 48.5%

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

       3. swap-sqr 48.5%

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

       4. unpow2 48.5%

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

   11. Simplified 48.5%

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

4. Final simplification 61.6%

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

Alternative 4: 55.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.u \leq 6.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, t\_0\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_0}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v)))
   (if (<= dY.u 6.5)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot (* (floor w) dX.u) t_0) (* (floor d) dX.w)) 2.0)
        (* (pow (floor d) 2.0) (pow dY.w 2.0)))))
     (log2
      (sqrt
       (fmax
        (pow t_0 2.0)
        (pow
         (hypot
          (* (floor d) dY.w)
          (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_u <= 6.5f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf((floorf(w) * dX_46_u), t_0), (floorf(d) * dX_46_w)), 2.0f), (powf(floorf(d), 2.0f) * powf(dY_46_w, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_0, 2.0f), powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(6.5))
		tmp = log2(sqrt((((hypot(hypot(Float32(floor(w) * dX_46_u), t_0), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) != (hypot(hypot(Float32(floor(w) * dX_46_u), t_0), Float32(floor(d) * dX_46_w)) ^ Float32(2.0))) ? Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) : ((Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) != Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0)))) ? (hypot(hypot(Float32(floor(w) * dX_46_u), t_0), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) : max((hypot(hypot(Float32(floor(w) * dX_46_u), t_0), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)), Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))))))));
	else
		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), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (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), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_u <= single(6.5))
		tmp = log2(sqrt(max((hypot(hypot((floor(w) * dX_46_u), t_0), (floor(d) * dX_46_w)) ^ single(2.0)), ((floor(d) ^ single(2.0)) * (dY_46_w ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((t_0 ^ single(2.0)), (hypot((floor(d) * dY_46_w), hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 6.5

   1. Initial program 66.2%

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

   3. Taylor expanded in w around 0 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in dY.u around 0 62.5%

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

      1. *-commutative 62.5%

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

   8. Simplified 62.5%

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

   9. Taylor expanded in dY.w around inf 56.0%

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

       1. *-commutative 56.0%

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

   11. Simplified 56.0%

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

   if 6.5 < dY.u

   1. Initial program 61.1%

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

   3. Taylor expanded in w around 0 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in dX.v around inf 56.1%

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

      1. unpow2 56.1%

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

      2. unpow2 56.1%

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

      3. swap-sqr 56.1%

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

      4. unpow2 56.1%

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

   8. Simplified 56.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 6.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ \mathbf{if}\;dY.u \leq 6.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, t\_1\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {t\_0}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_1}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

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)))
   (if (<= dY.u 6.5)
     (log2
      (sqrt
       (fmax
        (pow (hypot (hypot (* (floor w) dX.u) t_1) (* (floor d) dX.w)) 2.0)
        (pow t_0 2.0))))
     (log2
      (sqrt
       (fmax
        (pow t_1 2.0)
        (pow
         (hypot t_0 (hypot (* (floor w) dY.u) (* (floor h) dY.v)))
         2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dY_46_w;
	float t_1 = floorf(h) * dX_46_v;
	float tmp;
	if (dY_46_u <= 6.5f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf(hypotf((floorf(w) * dX_46_u), t_1), (floorf(d) * dX_46_w)), 2.0f), powf(t_0, 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(t_1, 2.0f), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w)
	t_1 = Float32(floor(h) * dX_46_v)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(6.5))
		tmp = log2(sqrt((((hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) != (hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0))) ? (t_0 ^ Float32(2.0)) : (((t_0 ^ Float32(2.0)) != (t_0 ^ Float32(2.0))) ? (hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) : max((hypot(hypot(Float32(floor(w) * dX_46_u), t_1), Float32(floor(d) * dX_46_w)) ^ 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(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) : (((hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0))) ? (t_1 ^ Float32(2.0)) : max((t_1 ^ Float32(2.0)), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), Float32(floor(h) * dY_46_v))) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dY_46_w;
	t_1 = floor(h) * dX_46_v;
	tmp = single(0.0);
	if (dY_46_u <= single(6.5))
		tmp = log2(sqrt(max((hypot(hypot((floor(w) * dX_46_u), t_1), (floor(d) * dX_46_w)) ^ single(2.0)), (t_0 ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((t_1 ^ single(2.0)), (hypot(t_0, hypot((floor(w) * dY_46_u), (floor(h) * dY_46_v))) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 6.5

   1. Initial program 66.2%

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

   3. Taylor expanded in w around 0 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in dY.u around 0 62.5%

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

      1. *-commutative 62.5%

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

   8. Simplified 62.5%

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

      1. add-exp-log 62.4%

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

      2. log-pow 62.4%

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

   10. Applied egg-rr 62.4%

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

   11. Taylor expanded in dY.w around inf 56.0%

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

       1. *-commutative 56.0%

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

       2. unpow2 56.0%

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

       3. unpow2 56.0%

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

       4. swap-sqr 56.0%

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

       5. unpow2 56.0%

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

   13. Simplified 56.0%

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

   if 6.5 < dY.u

   1. Initial program 61.1%

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

   3. Taylor expanded in w around 0 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in dX.v around inf 56.1%

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

      1. unpow2 56.1%

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

      2. unpow2 56.1%

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

      3. swap-sqr 56.1%

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

      4. unpow2 56.1%

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

   8. Simplified 56.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 6.5:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot dY.w\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_1 := {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\\ \mathbf{if}\;dX.u \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor h\right\rfloor \cdot dX.v, t\_0\right)\right)}^{2}, t\_1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, t\_0\right)\right)}^{2}, t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dX.w))
        (t_1 (pow (hypot (* (floor d) dY.w) (* (floor h) dY.v)) 2.0)))
   (if (<= dX.u 180.0)
     (log2 (sqrt (fmax (pow (hypot (* (floor h) dX.v) t_0) 2.0) t_1)))
     (log2 (sqrt (fmax (pow (hypot (* (floor w) dX.u) t_0) 2.0) t_1))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dX_46_w;
	float t_1 = powf(hypotf((floorf(d) * dY_46_w), (floorf(h) * dY_46_v)), 2.0f);
	float tmp;
	if (dX_46_u <= 180.0f) {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(h) * dX_46_v), t_0), 2.0f), t_1)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), t_0), 2.0f), t_1)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dX_46_w)
	t_1 = hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(180.0))
		tmp = log2(sqrt((((hypot(Float32(floor(h) * dX_46_v), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(h) * dX_46_v), t_0) ^ Float32(2.0))) ? t_1 : ((t_1 != t_1) ? (hypot(Float32(floor(h) * dX_46_v), t_0) ^ Float32(2.0)) : max((hypot(Float32(floor(h) * dX_46_v), t_0) ^ Float32(2.0)), t_1)))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0))) ? t_1 : ((t_1 != t_1) ? (hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), t_0) ^ Float32(2.0)), t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = floor(d) * dX_46_w;
	t_1 = hypot((floor(d) * dY_46_w), (floor(h) * dY_46_v)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(180.0))
		tmp = log2(sqrt(max((hypot((floor(h) * dX_46_v), t_0) ^ single(2.0)), t_1)));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), t_0) ^ single(2.0)), t_1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 180

   1. Initial program 67.7%

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

   3. Taylor expanded in w around 0 67.7%

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

   5. Simplified 67.7%

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

   6. Taylor expanded in dY.u around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in dX.u around 0 57.3%

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

   if 180 < dX.u

   1. Initial program 53.2%

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

   3. Taylor expanded in w around 0 53.2%

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

   5. Simplified 53.2%

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

   6. Taylor expanded in dY.u around 0 51.5%

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

      1. *-commutative 51.5%

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

   8. Simplified 51.5%

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

   9. Taylor expanded in dX.u around inf 51.0%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor h\right\rfloor \cdot dX.v, \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.w \leq 0.004000000189989805:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_0\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, t\_0\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v)) (t_1 (* (floor d) dY.w)))
   (if (<= dX.w 0.004000000189989805)
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (pow (hypot t_1 (hypot (* (floor w) dY.u) t_0)) 2.0))))
     (log2
      (sqrt
       (fmax
        (pow (hypot (* (floor w) dX.u) (* (floor d) dX.w)) 2.0)
        (pow (hypot t_1 t_0) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_w <= 0.004000000189989805f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), t_0)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf(hypotf((floorf(w) * dX_46_u), (floorf(d) * dX_46_w)), 2.0f), powf(hypotf(t_1, t_0), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(0.004000000189989805))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) : (((hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt((((hypot(Float32(floor(w) * dX_46_u), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) != (hypot(Float32(floor(w) * dX_46_u), Float32(floor(d) * dX_46_w)) ^ Float32(2.0))) ? (hypot(t_1, t_0) ^ Float32(2.0)) : (((hypot(t_1, t_0) ^ Float32(2.0)) != (hypot(t_1, t_0) ^ Float32(2.0))) ? (hypot(Float32(floor(w) * dX_46_u), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)) : max((hypot(Float32(floor(w) * dX_46_u), Float32(floor(d) * dX_46_w)) ^ Float32(2.0)), (hypot(t_1, 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;
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_w <= single(0.004000000189989805))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), (hypot(t_1, hypot((floor(w) * dY_46_u), t_0)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max((hypot((floor(w) * dX_46_u), (floor(d) * dX_46_w)) ^ single(2.0)), (hypot(t_1, t_0) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 0.00400000019

   1. Initial program 66.3%

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

   3. Taylor expanded in w around 0 66.3%

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

   5. Simplified 66.3%

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

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

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

      1. unpow2 54.2%

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

      2. unpow2 54.2%

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

      3. swap-sqr 54.2%

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

      4. unpow2 54.2%

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

   8. Simplified 54.2%

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

   if 0.00400000019 < dX.w

   1. Initial program 61.3%

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

   3. Taylor expanded in w around 0 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in dY.u around 0 58.2%

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

      1. *-commutative 58.2%

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

   8. Simplified 58.2%

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

   9. Taylor expanded in dX.u around inf 57.5%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 0.004000000189989805:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\\ \mathbf{if}\;dX.w \leq 0.004000000189989805:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

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.w 0.004000000189989805)
     (log2 (sqrt (fmax (pow (* (floor h) dX.v) 2.0) t_0)))
     (log2 (sqrt (fmax (pow (* (floor d) dX.w) 2.0) t_0))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf(hypotf((floorf(d) * dY_46_w), hypotf((floorf(w) * dY_46_u), (floorf(h) * dY_46_v))), 2.0f);
	float tmp;
	if (dX_46_w <= 0.004000000189989805f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(d) * dX_46_w), 2.0f), t_0)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = 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_w <= Float32(0.004000000189989805))
		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(d) * dX_46_w) ^ Float32(2.0)) != (Float32(floor(d) * dX_46_w) ^ Float32(2.0))) ? t_0 : ((t_0 != t_0) ? (Float32(floor(d) * dX_46_w) ^ Float32(2.0)) : max((Float32(floor(d) * dX_46_w) ^ Float32(2.0)), t_0)))));
	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_w <= single(0.004000000189989805))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((floor(d) * dX_46_w) ^ single(2.0)), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 0.00400000019

   1. Initial program 66.3%

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

   3. Taylor expanded in w around 0 66.3%

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

   5. Simplified 66.3%

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

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

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

      1. unpow2 54.2%

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

      2. unpow2 54.2%

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

      3. swap-sqr 54.2%

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

      4. unpow2 54.2%

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

   8. Simplified 54.2%

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

   if 0.00400000019 < dX.w

   1. Initial program 61.3%

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

   3. Taylor expanded in w around 0 61.3%

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

   5. Simplified 61.3%

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

   6. Taylor expanded in dX.v around 0 60.7%

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

      1. +-commutative 60.7%

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

      2. fma-define 60.7%

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

      3. unpow2 60.7%

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

      4. unpow2 60.7%

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

      5. swap-sqr 60.7%

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

      6. unpow2 60.7%

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

   8. Simplified 60.7%

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

   9. Taylor expanded in dX.w around inf 56.2%

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

       1. unpow2 56.2%

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

       2. unpow2 56.2%

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

       3. swap-sqr 56.2%

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

       4. unpow2 56.2%

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

   11. Simplified 56.2%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 0.004000000189989805:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor d\right\rfloor \cdot dX.w\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.w \leq 3000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_0\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(\mathsf{hypot}\left(t\_1, t\_0\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v)) (t_1 (* (floor d) dY.w)))
   (if (<= dX.w 3000000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor h) dX.v) 2.0)
        (pow (hypot t_1 (hypot (* (floor w) dY.u) t_0)) 2.0))))
     (log2
      (sqrt
       (fmax
        (* (pow dX.w 2.0) (pow (floor d) 2.0))
        (pow (hypot t_1 t_0) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v;
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_w <= 3000000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), powf(hypotf(t_1, hypotf((floorf(w) * dY_46_u), t_0)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_w, 2.0f) * powf(floorf(d), 2.0f)), powf(hypotf(t_1, t_0), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(3000000000.0))
		tmp = log2(sqrt((((Float32(floor(h) * dX_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dX_46_v) ^ Float32(2.0))) ? (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) : (((hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)) != (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0))) ? (Float32(floor(h) * dX_46_v) ^ Float32(2.0)) : max((Float32(floor(h) * dX_46_v) ^ Float32(2.0)), (hypot(t_1, hypot(Float32(floor(w) * dY_46_u), t_0)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))) != Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0)))) ? (hypot(t_1, t_0) ^ Float32(2.0)) : (((hypot(t_1, t_0) ^ Float32(2.0)) != (hypot(t_1, t_0) ^ Float32(2.0))) ? Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))) : max(Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))), (hypot(t_1, 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;
	t_1 = floor(d) * dY_46_w;
	tmp = single(0.0);
	if (dX_46_w <= single(3000000000.0))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), (hypot(t_1, hypot((floor(w) * dY_46_u), t_0)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((dX_46_w ^ single(2.0)) * (floor(d) ^ single(2.0))), (hypot(t_1, t_0) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 3e9

   1. Initial program 67.5%

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

   3. Taylor expanded in w around 0 67.5%

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

   5. Simplified 67.5%

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

   6. Taylor expanded in dX.v around inf 54.7%

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

      1. unpow2 54.7%

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

      2. unpow2 54.7%

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

      3. swap-sqr 54.7%

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

      4. unpow2 54.7%

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

   8. Simplified 54.7%

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

   if 3e9 < dX.w

   1. Initial program 45.7%

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

   3. Taylor expanded in w around 0 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in dY.u around 0 48.7%

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

      1. *-commutative 48.7%

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

   8. Simplified 48.7%

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

   9. Taylor expanded in dX.w around inf 51.0%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 3000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\\ \mathbf{if}\;dX.w \leq 500000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, t\_1\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(\mathsf{hypot}\left(t\_0, t\_1\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* (floor d) dY.w)) (t_1 (* (floor h) dY.v)))
   (if (<= dX.w 500000000.0)
     (log2
      (sqrt
       (fmax
        (pow (* (floor w) dX.u) 2.0)
        (pow (hypot t_0 (hypot (* (floor w) dY.u) t_1)) 2.0))))
     (log2
      (sqrt
       (fmax
        (* (pow dX.w 2.0) (pow (floor d) 2.0))
        (pow (hypot t_0 t_1) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(d) * dY_46_w;
	float t_1 = floorf(h) * dY_46_v;
	float tmp;
	if (dX_46_w <= 500000000.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(w) * dX_46_u), 2.0f), powf(hypotf(t_0, hypotf((floorf(w) * dY_46_u), t_1)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_w, 2.0f) * powf(floorf(d), 2.0f)), powf(hypotf(t_0, t_1), 2.0f))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(d) * dY_46_w)
	t_1 = Float32(floor(h) * dY_46_v)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(500000000.0))
		tmp = log2(sqrt((((Float32(floor(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) : (((hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)) != (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), (hypot(t_0, hypot(Float32(floor(w) * dY_46_u), t_1)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))) != Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0)))) ? (hypot(t_0, t_1) ^ Float32(2.0)) : (((hypot(t_0, t_1) ^ Float32(2.0)) != (hypot(t_0, t_1) ^ Float32(2.0))) ? Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))) : max(Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))), (hypot(t_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(d) * dY_46_w;
	t_1 = floor(h) * dY_46_v;
	tmp = single(0.0);
	if (dX_46_w <= single(500000000.0))
		tmp = log2(sqrt(max(((floor(w) * dX_46_u) ^ single(2.0)), (hypot(t_0, hypot((floor(w) * dY_46_u), t_1)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(((dX_46_w ^ single(2.0)) * (floor(d) ^ single(2.0))), (hypot(t_0, t_1) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 5e8

   1. Initial program 67.4%

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

   3. Taylor expanded in w around 0 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in dX.u around inf 52.4%

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

      1. unpow2 45.2%

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

      2. unpow2 45.2%

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

      3. swap-sqr 45.2%

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

      4. unpow2 45.2%

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

   8. Simplified 52.4%

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

   if 5e8 < dX.w

   1. Initial program 47.4%

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

   3. Taylor expanded in w around 0 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in dY.u around 0 48.1%

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

      1. *-commutative 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in dX.w around inf 50.4%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 500000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \mathsf{hypot}\left(\left\lfloor w\right\rfloor \cdot dY.u, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\\ \mathbf{if}\;dX.w \leq 0.019999999552965164:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (hypot (* (floor d) dY.w) (* (floor h) dY.v)) 2.0)))
   (if (<= dX.w 0.019999999552965164)
     (log2 (sqrt (fmax (pow (* (floor h) dX.v) 2.0) t_0)))
     (log2 (sqrt (fmax (* (pow dX.w 2.0) (pow (floor d) 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), (floorf(h) * dY_46_v)), 2.0f);
	float tmp;
	if (dX_46_w <= 0.019999999552965164f) {
		tmp = log2f(sqrtf(fmaxf(powf((floorf(h) * dX_46_v), 2.0f), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(dX_46_w, 2.0f) * powf(floorf(d), 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), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(0.019999999552965164))
		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((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))) != Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0)))) ? t_0 : ((t_0 != t_0) ? Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ Float32(2.0))) : max(Float32((dX_46_w ^ Float32(2.0)) * (floor(d) ^ 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), (floor(h) * dY_46_v)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_w <= single(0.019999999552965164))
		tmp = log2(sqrt(max(((floor(h) * dX_46_v) ^ single(2.0)), t_0)));
	else
		tmp = log2(sqrt(max(((dX_46_w ^ single(2.0)) * (floor(d) ^ single(2.0))), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 0.0199999996

   1. Initial program 66.6%

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

   3. Taylor expanded in w around 0 66.6%

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

   5. Simplified 66.6%

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

   6. Taylor expanded in dY.u around 0 60.7%

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

      1. *-commutative 60.7%

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

   8. Simplified 60.7%

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

   9. Taylor expanded in dX.v around inf 48.0%

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

       1. unpow2 54.7%

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

       2. unpow2 54.7%

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

       3. swap-sqr 54.7%

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

       4. unpow2 54.7%

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

   11. Simplified 48.0%

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

   if 0.0199999996 < dX.w

   1. Initial program 60.1%

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

   3. Taylor expanded in w around 0 60.1%

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

   5. Simplified 60.1%

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

   6. Taylor expanded in dY.u around 0 56.9%

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

      1. *-commutative 56.9%

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

   8. Simplified 56.9%

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

   9. Taylor expanded in dX.w around inf 51.1%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.w \leq 0.019999999552965164:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({dX.w}^{2} \cdot {\left(\left\lfloor d\right\rfloor \right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\\ \mathbf{if}\;dX.u \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, t\_0\right)}\right)\\ \end{array} \end{array} \]

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) (* (floor h) dY.v)) 2.0)))
   (if (<= dX.u 180.0)
     (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), (floorf(h) * dY_46_v)), 2.0f);
	float tmp;
	if (dX_46_u <= 180.0f) {
		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), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(180.0))
		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), (floor(h) * dY_46_v)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u <= single(180.0))
		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 < 180

   1. Initial program 67.7%

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

   3. Taylor expanded in w around 0 67.7%

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

   5. Simplified 67.7%

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

   6. Taylor expanded in dY.u around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in dX.v around inf 47.4%

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

       1. unpow2 54.8%

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

       2. unpow2 54.8%

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

       3. swap-sqr 54.8%

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

       4. unpow2 54.8%

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

   11. Simplified 47.4%

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

   if 180 < dX.u

   1. Initial program 53.2%

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

   3. Taylor expanded in w around 0 53.2%

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

   5. Simplified 53.2%

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

   6. Taylor expanded in dY.u around 0 51.5%

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

      1. *-commutative 51.5%

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

   8. Simplified 51.5%

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

   9. Taylor expanded in dX.u around inf 45.0%

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

       1. unpow2 45.0%

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

       2. unpow2 45.0%

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

       3. swap-sqr 45.0%

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

       4. unpow2 45.0%

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

   11. Simplified 45.0%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 180:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor h\right\rfloor \cdot dX.v\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dY.v \leq 25:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\mathsf{hypot}\left(t\_1, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\mathsf{hypot}\left(t\_1, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* (floor w) dX.u) 2.0)) (t_1 (* (floor d) dY.w)))
   (if (<= dY.v 25.0)
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (floor w) dY.u)) 2.0))))
     (log2 (sqrt (fmax t_0 (pow (hypot t_1 (* (floor h) dY.v)) 2.0)))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((floorf(w) * dX_46_u), 2.0f);
	float t_1 = floorf(d) * dY_46_w;
	float tmp;
	if (dY_46_v <= 25.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (floorf(w) * dY_46_u)), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf(hypotf(t_1, (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(w) * dX_46_u) ^ Float32(2.0)
	t_1 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(25.0))
		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)))))));
	else
		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)))))));
	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_v <= single(25.0))
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(w) * dY_46_u)) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, (hypot(t_1, (floor(h) * dY_46_v)) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 25

   1. Initial program 66.7%

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

   3. Taylor expanded in w around 0 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in dX.u around inf 48.0%

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

      1. unpow2 40.3%

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

      2. unpow2 40.3%

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

      3. swap-sqr 40.3%

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

      4. unpow2 40.3%

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

   8. Simplified 48.0%

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

   9. Taylor expanded in dY.u around inf 44.3%

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

       1. *-commutative 44.3%

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

   11. Simplified 44.3%

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

   if 25 < dY.v

   1. Initial program 59.3%

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

   3. Taylor expanded in w around 0 59.3%

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

   5. Simplified 59.3%

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

   6. Taylor expanded in dY.u around 0 57.4%

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

      1. *-commutative 57.4%

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

   8. Simplified 57.4%

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

   9. Taylor expanded in dX.u around inf 50.1%

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

       1. unpow2 50.1%

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

       2. unpow2 50.1%

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

       3. swap-sqr 50.1%

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

       4. unpow2 50.1%

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

   11. Simplified 50.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 25:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor w\right\rfloor \cdot dY.u\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\mathsf{hypot}\left(\left\lfloor d\right\rfloor \cdot dY.w, \left\lfloor h\right\rfloor \cdot dY.v\right)\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.5% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax
    (pow (* (floor w) dX.u) 2.0)
    (pow (hypot (* (floor d) dY.w) (* (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(w) * dX_46_u), 2.0f), powf(hypotf((floorf(d) * dY_46_w), (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(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ Float32(2.0))) ? (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) : (((hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0)) != (hypot(Float32(floor(d) * dY_46_w), Float32(floor(h) * dY_46_v)) ^ Float32(2.0))) ? (Float32(floor(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ Float32(2.0)), (hypot(Float32(floor(d) * dY_46_w), 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(w) * dX_46_u) ^ single(2.0)), (hypot((floor(d) * dY_46_w), (floor(h) * dY_46_v)) ^ single(2.0)))));
end

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in w around 0 64.9%

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

5. Simplified 64.9%

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

6. Taylor expanded in dY.u around 0 59.7%

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

   1. *-commutative 59.7%

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

8. Simplified 59.7%

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

9. Taylor expanded in dX.u around inf 42.7%

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

    1. unpow2 42.7%

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

    2. unpow2 42.7%

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

    3. swap-sqr 42.7%

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

    4. unpow2 42.7%

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

11. Simplified 42.7%

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

12. Final simplification 42.7%

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

Alternative 15: 38.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ \mathbf{if}\;dY.v \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

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)))
   (if (<= dY.v 500000.0)
     (log2 (sqrt (fmax t_0 (* (pow (floor d) 2.0) (pow dY.w 2.0)))))
     (log2 (sqrt (fmax t_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) {
	float t_0 = powf((floorf(w) * dX_46_u), 2.0f);
	float tmp;
	if (dY_46_v <= 500000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, (powf(floorf(d), 2.0f) * powf(dY_46_w, 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((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(w) * dX_46_u) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(500000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) : ((Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))) != Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0)))) ? t_0 : max(t_0, Float32((floor(d) ^ Float32(2.0)) * (dY_46_w ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(h) * dY_46_v) ^ Float32(2.0)) : (((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	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);
	tmp = single(0.0);
	if (dY_46_v <= single(500000.0))
		tmp = log2(sqrt(max(t_0, ((floor(d) ^ single(2.0)) * (dY_46_w ^ single(2.0))))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(h) * dY_46_v) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 5e5

   1. Initial program 65.9%

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

   3. Taylor expanded in w around 0 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in dY.u around 0 60.0%

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

      1. *-commutative 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in dX.u around inf 40.5%

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

       1. unpow2 40.5%

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

       2. unpow2 40.5%

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

       3. swap-sqr 40.5%

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

       4. unpow2 40.5%

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

   11. Simplified 40.5%

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

   12. Taylor expanded in dY.w around inf 35.7%

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

       1. *-commutative 56.4%

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

   14. Simplified 35.7%

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

   if 5e5 < dY.v

   1. Initial program 61.2%

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

   3. Taylor expanded in w around 0 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in dY.u around 0 58.6%

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

      1. *-commutative 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in dX.u around inf 51.0%

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

       1. unpow2 51.0%

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

       2. unpow2 51.0%

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

       3. swap-sqr 51.0%

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

       4. unpow2 51.0%

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

   11. Simplified 51.0%

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

   12. Taylor expanded in dY.w around 0 51.7%

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

       1. *-commutative 51.7%

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

   14. Simplified 51.7%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot {dY.w}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}\\ \mathbf{if}\;dY.v \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor d\right\rfloor \cdot \left(-dY.w\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

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)))
   (if (<= dY.v 500000.0)
     (log2 (sqrt (fmax t_0 (pow (* (floor d) (- dY.w)) 2.0))))
     (log2 (sqrt (fmax t_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) {
	float t_0 = powf((floorf(w) * dX_46_u), 2.0f);
	float tmp;
	if (dY_46_v <= 500000.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((floorf(d) * -dY_46_w), 2.0f))));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_0, powf((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(w) * dX_46_u) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dY_46_v <= Float32(500000.0))
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(d) * Float32(-dY_46_w)) ^ Float32(2.0)) : (((Float32(floor(d) * Float32(-dY_46_w)) ^ Float32(2.0)) != (Float32(floor(d) * Float32(-dY_46_w)) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(d) * Float32(-dY_46_w)) ^ Float32(2.0)))))));
	else
		tmp = log2(sqrt(((t_0 != t_0) ? (Float32(floor(h) * dY_46_v) ^ Float32(2.0)) : (((Float32(floor(h) * dY_46_v) ^ Float32(2.0)) != (Float32(floor(h) * dY_46_v) ^ Float32(2.0))) ? t_0 : max(t_0, (Float32(floor(h) * dY_46_v) ^ Float32(2.0)))))));
	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);
	tmp = single(0.0);
	if (dY_46_v <= single(500000.0))
		tmp = log2(sqrt(max(t_0, ((floor(d) * -dY_46_w) ^ single(2.0)))));
	else
		tmp = log2(sqrt(max(t_0, ((floor(h) * dY_46_v) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 5e5

   1. Initial program 65.9%

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

   3. Taylor expanded in w around 0 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in dY.u around 0 60.0%

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

      1. *-commutative 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in dX.u around inf 40.5%

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

       1. unpow2 40.5%

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

       2. unpow2 40.5%

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

       3. swap-sqr 40.5%

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

       4. unpow2 40.5%

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

   11. Simplified 40.5%

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

   12. Taylor expanded in dY.w around -inf 35.7%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, {\color{blue}{\left(-1 \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right)\right)}}^{2}\right)}\right) \]
   13. Step-by-step derivation

       1. mul-1-neg 35.7%

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

       2. *-commutative 35.7%

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

       3. distribute-rgt-neg-in 35.7%

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

   14. Simplified 35.7%

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

   if 5e5 < dY.v

   1. Initial program 61.2%

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

   3. Taylor expanded in w around 0 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in dY.u around 0 58.6%

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

      1. *-commutative 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in dX.u around inf 51.0%

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

       1. unpow2 51.0%

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

       2. unpow2 51.0%

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

       3. swap-sqr 51.0%

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

       4. unpow2 51.0%

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

   11. Simplified 51.0%

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

   12. Taylor expanded in dY.w around 0 51.7%

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

       1. *-commutative 51.7%

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

   14. Simplified 51.7%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.v \leq 500000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor d\right\rfloor \cdot \left(-dY.w\right)\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(\left\lfloor w\right\rfloor \cdot dX.u\right)}^{2}, {\left(\left\lfloor h\right\rfloor \cdot dY.v\right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.1% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt (fmax (pow (* (floor w) dX.u) 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(w) * dX_46_u), 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(w) * dX_46_u) ^ Float32(2.0)) != (Float32(floor(w) * dX_46_u) ^ 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(w) * dX_46_u) ^ Float32(2.0)) : max((Float32(floor(w) * dX_46_u) ^ 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(w) * dX_46_u) ^ single(2.0)), ((floor(h) * dY_46_v) ^ single(2.0)))));
end

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in w around 0 64.9%

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

5. Simplified 64.9%

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

6. Taylor expanded in dY.u around 0 59.7%

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

   1. *-commutative 59.7%

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

8. Simplified 59.7%

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

9. Taylor expanded in dX.u around inf 42.7%

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

    1. unpow2 42.7%

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

    2. unpow2 42.7%

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

    3. swap-sqr 42.7%

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

    4. unpow2 42.7%

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

11. Simplified 42.7%

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

12. Taylor expanded in dY.w around 0 33.8%

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

    1. *-commutative 33.8%

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

14. Simplified 33.8%

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

15. Final simplification 33.8%

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

Reproduce

herbie shell --seed 2024163 
(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)))))))