Isotropic LOD (LOD)

Percentage Accurate: 68.6% → 71.5%

Time: 22.8s

Alternatives: 11

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: 68.6% 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: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_2 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_3 := dX.u\_m \cdot \left\lfloor w\right\rfloor \\ t_4 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_5 := dY.w \cdot \left\lfloor d\right\rfloor \\ t_6 := \mathsf{max}\left(t\_4 \cdot t\_4 + \left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right), t\_5 \cdot t\_5 + \left(t\_0 \cdot t\_0 + t\_2 \cdot t\_2\right)\right)\\ \mathbf{if}\;t\_6 \leq 2.500000072189495 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{t\_6}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_4}^{2} + \left({t\_3}^{2} + {t\_1}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h)))
        (t_1 (* dX.v (floor h)))
        (t_2 (* dY.u (floor w)))
        (t_3 (* dX.u_m (floor w)))
        (t_4 (* dX.w (floor d)))
        (t_5 (* dY.w (floor d)))
        (t_6
         (fmax
          (+ (* t_4 t_4) (+ (* t_1 t_1) (* t_3 t_3)))
          (+ (* t_5 t_5) (+ (* t_0 t_0) (* t_2 t_2))))))
   (if (<= t_6 2.500000072189495e+38)
     (log2 (sqrt t_6))
     (log2
      (sqrt
       (fmax
        (+ (pow t_4 2.0) (+ (pow t_3 2.0) (pow t_1 2.0)))
        (fma
         (* (pow (floor d) 2.0) dY.w)
         dY.w
         (* (* (pow (floor w) 2.0) dY.u) dY.u))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = dX_46_v * floorf(h);
	float t_2 = dY_46_u * floorf(w);
	float t_3 = dX_46_u_m * floorf(w);
	float t_4 = dX_46_w * floorf(d);
	float t_5 = dY_46_w * floorf(d);
	float t_6 = fmaxf(((t_4 * t_4) + ((t_1 * t_1) + (t_3 * t_3))), ((t_5 * t_5) + ((t_0 * t_0) + (t_2 * t_2))));
	float tmp;
	if (t_6 <= 2.500000072189495e+38f) {
		tmp = log2f(sqrtf(t_6));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(t_4, 2.0f) + (powf(t_3, 2.0f) + powf(t_1, 2.0f))), fmaf((powf(floorf(d), 2.0f) * dY_46_w), dY_46_w, ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u)))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dX_46_v * floor(h))
	t_2 = Float32(dY_46_u * floor(w))
	t_3 = Float32(dX_46_u_m * floor(w))
	t_4 = Float32(dX_46_w * floor(d))
	t_5 = Float32(dY_46_w * floor(d))
	t_6 = (Float32(Float32(t_4 * t_4) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3))) != Float32(Float32(t_4 * t_4) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)))) ? Float32(Float32(t_5 * t_5) + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2))) : ((Float32(Float32(t_5 * t_5) + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2))) != Float32(Float32(t_5 * t_5) + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)))) ? Float32(Float32(t_4 * t_4) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3))) : max(Float32(Float32(t_4 * t_4) + Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3))), Float32(Float32(t_5 * t_5) + Float32(Float32(t_0 * t_0) + Float32(t_2 * t_2)))))
	tmp = Float32(0.0)
	if (t_6 <= Float32(2.500000072189495e+38))
		tmp = log2(sqrt(t_6));
	else
		tmp = log2(sqrt(((Float32((t_4 ^ Float32(2.0)) + Float32((t_3 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) != Float32((t_4 ^ Float32(2.0)) + Float32((t_3 ^ Float32(2.0)) + (t_1 ^ Float32(2.0))))) ? fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)) : ((fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)) != fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u))) ? Float32((t_4 ^ Float32(2.0)) + Float32((t_3 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))) : max(Float32((t_4 ^ Float32(2.0)) + Float32((t_3 ^ Float32(2.0)) + (t_1 ^ Float32(2.0)))), fma(Float32((floor(d) ^ Float32(2.0)) * dY_46_w), dY_46_w, Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))) < 2.5000001e38

   1. Initial program 100.0%

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

   if 2.5000001e38 < (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))

   1. Initial program 7.5%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 16.6

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

   5. Applied rewrites 16.6%

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

   7. Applied rewrites 16.6%

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

   8. Taylor expanded in dY.v around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-floor.f32 N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-pow.f32 N/A

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

      17. lower-floor.f32 20.3

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

   10. Applied rewrites 20.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{max}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right), \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)\right) \leq 2.500000072189495 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right), \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \mathsf{fma}\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dY.w, dY.w, \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_1 := t\_0 \cdot t\_0\\ t_2 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_3 := {\left(\left\lfloor h\right\rfloor \right)}^{2}\\ t_4 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_5 := dX.u\_m \cdot \left\lfloor w\right\rfloor \\ t_6 := dX.v \cdot \left\lfloor h\right\rfloor \\ t_7 := dY.w \cdot \left\lfloor d\right\rfloor \\ t_8 := t\_7 \cdot t\_7\\ \mathbf{if}\;dY.u \leq 23000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_1 + \left(t\_6 \cdot t\_6 + t\_5 \cdot t\_5\right), \left(t\_3 \cdot dY.v\right) \cdot dY.v + t\_8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_3 \cdot dX.v\right) \cdot dX.v + t\_1, t\_8 + \left(t\_2 \cdot t\_2 + t\_4 \cdot t\_4\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dX.w (floor d)))
        (t_1 (* t_0 t_0))
        (t_2 (* dY.v (floor h)))
        (t_3 (pow (floor h) 2.0))
        (t_4 (* dY.u (floor w)))
        (t_5 (* dX.u_m (floor w)))
        (t_6 (* dX.v (floor h)))
        (t_7 (* dY.w (floor d)))
        (t_8 (* t_7 t_7)))
   (if (<= dY.u 23000000.0)
     (log2
      (sqrt
       (fmax
        (+ t_1 (+ (* t_6 t_6) (* t_5 t_5)))
        (+ (* (* t_3 dY.v) dY.v) t_8))))
     (log2
      (sqrt
       (fmax
        (+ (* (* t_3 dX.v) dX.v) t_1)
        (+ t_8 (+ (* t_2 t_2) (* t_4 t_4)))))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = dX_46_w * floorf(d);
	float t_1 = t_0 * t_0;
	float t_2 = dY_46_v * floorf(h);
	float t_3 = powf(floorf(h), 2.0f);
	float t_4 = dY_46_u * floorf(w);
	float t_5 = dX_46_u_m * floorf(w);
	float t_6 = dX_46_v * floorf(h);
	float t_7 = dY_46_w * floorf(d);
	float t_8 = t_7 * t_7;
	float tmp;
	if (dY_46_u <= 23000000.0f) {
		tmp = log2f(sqrtf(fmaxf((t_1 + ((t_6 * t_6) + (t_5 * t_5))), (((t_3 * dY_46_v) * dY_46_v) + t_8))));
	} else {
		tmp = log2f(sqrtf(fmaxf((((t_3 * dX_46_v) * dX_46_v) + t_1), (t_8 + ((t_2 * t_2) + (t_4 * t_4))))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w * floor(d))
	t_1 = Float32(t_0 * t_0)
	t_2 = Float32(dY_46_v * floor(h))
	t_3 = floor(h) ^ Float32(2.0)
	t_4 = Float32(dY_46_u * floor(w))
	t_5 = Float32(dX_46_u_m * floor(w))
	t_6 = Float32(dX_46_v * floor(h))
	t_7 = Float32(dY_46_w * floor(d))
	t_8 = Float32(t_7 * t_7)
	tmp = Float32(0.0)
	if (dY_46_u <= Float32(23000000.0))
		tmp = log2(sqrt(((Float32(t_1 + Float32(Float32(t_6 * t_6) + Float32(t_5 * t_5))) != Float32(t_1 + Float32(Float32(t_6 * t_6) + Float32(t_5 * t_5)))) ? Float32(Float32(Float32(t_3 * dY_46_v) * dY_46_v) + t_8) : ((Float32(Float32(Float32(t_3 * dY_46_v) * dY_46_v) + t_8) != Float32(Float32(Float32(t_3 * dY_46_v) * dY_46_v) + t_8)) ? Float32(t_1 + Float32(Float32(t_6 * t_6) + Float32(t_5 * t_5))) : max(Float32(t_1 + Float32(Float32(t_6 * t_6) + Float32(t_5 * t_5))), Float32(Float32(Float32(t_3 * dY_46_v) * dY_46_v) + t_8))))));
	else
		tmp = log2(sqrt(((Float32(Float32(Float32(t_3 * dX_46_v) * dX_46_v) + t_1) != Float32(Float32(Float32(t_3 * dX_46_v) * dX_46_v) + t_1)) ? Float32(t_8 + Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4))) : ((Float32(t_8 + Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4))) != Float32(t_8 + Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4)))) ? Float32(Float32(Float32(t_3 * dX_46_v) * dX_46_v) + t_1) : max(Float32(Float32(Float32(t_3 * dX_46_v) * dX_46_v) + t_1), Float32(t_8 + Float32(Float32(t_2 * t_2) + Float32(t_4 * t_4))))))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = dX_46_w * floor(d);
	t_1 = t_0 * t_0;
	t_2 = dY_46_v * floor(h);
	t_3 = floor(h) ^ single(2.0);
	t_4 = dY_46_u * floor(w);
	t_5 = dX_46_u_m * floor(w);
	t_6 = dX_46_v * floor(h);
	t_7 = dY_46_w * floor(d);
	t_8 = t_7 * t_7;
	tmp = single(0.0);
	if (dY_46_u <= single(23000000.0))
		tmp = log2(sqrt(max((t_1 + ((t_6 * t_6) + (t_5 * t_5))), (((t_3 * dY_46_v) * dY_46_v) + t_8))));
	else
		tmp = log2(sqrt(max((((t_3 * dX_46_v) * dX_46_v) + t_1), (t_8 + ((t_2 * t_2) + (t_4 * t_4))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 2.3e7

   1. Initial program 71.6%

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

   3. Taylor expanded in dY.u around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 67.4

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

   5. Applied rewrites 67.4%

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

   if 2.3e7 < dY.u

   1. Initial program 49.2%

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

   3. Taylor expanded in dX.u around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 49.0

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

   5. Applied rewrites 49.0%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dY.u \leq 23000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dX.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dX.v \cdot \left\lfloor h\right\rfloor \right) + \left(dX.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor \right)\right), \left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dY.v\right) \cdot dY.v + \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v + \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right), \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := dY.v \cdot \left\lfloor h\right\rfloor \\ t_1 := dY.u \cdot \left\lfloor w\right\rfloor \\ t_2 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_3 := dY.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dX.u\_m \leq 1900:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v + t\_2 \cdot t\_2, t\_3 \cdot t\_3 + \left(t\_0 \cdot t\_0 + t\_1 \cdot t\_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({t\_2}^{2} + \left({\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (* dY.v (floor h)))
        (t_1 (* dY.u (floor w)))
        (t_2 (* dX.w (floor d)))
        (t_3 (* dY.w (floor d))))
   (if (<= dX.u_m 1900.0)
     (log2
      (sqrt
       (fmax
        (+ (* (* (pow (floor h) 2.0) dX.v) dX.v) (* t_2 t_2))
        (+ (* t_3 t_3) (+ (* t_0 t_0) (* t_1 t_1))))))
     (log2
      (sqrt
       (fmax
        (+
         (pow t_2 2.0)
         (+ (pow (* dX.u_m (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        (* (* (pow (floor w) 2.0) dY.u) dY.u)))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = dY_46_v * floorf(h);
	float t_1 = dY_46_u * floorf(w);
	float t_2 = dX_46_w * floorf(d);
	float t_3 = dY_46_w * floorf(d);
	float tmp;
	if (dX_46_u_m <= 1900.0f) {
		tmp = log2f(sqrtf(fmaxf((((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v) + (t_2 * t_2)), ((t_3 * t_3) + ((t_0 * t_0) + (t_1 * t_1))))));
	} else {
		tmp = log2f(sqrtf(fmaxf((powf(t_2, 2.0f) + (powf((dX_46_u_m * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h))
	t_1 = Float32(dY_46_u * floor(w))
	t_2 = Float32(dX_46_w * floor(d))
	t_3 = Float32(dY_46_w * floor(d))
	tmp = Float32(0.0)
	if (dX_46_u_m <= Float32(1900.0))
		tmp = log2(sqrt(((Float32(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) + Float32(t_2 * t_2)) != Float32(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) + Float32(t_2 * t_2))) ? Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))) : ((Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))) != Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)))) ? Float32(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) + Float32(t_2 * t_2)) : max(Float32(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) + Float32(t_2 * t_2)), Float32(Float32(t_3 * t_3) + Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1))))))));
	else
		tmp = log2(sqrt(((Float32((t_2 ^ Float32(2.0)) + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) != Float32((t_2 ^ Float32(2.0)) + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))))) ? Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) : ((Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) != Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)) ? Float32((t_2 ^ Float32(2.0)) + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) : max(Float32((t_2 ^ Float32(2.0)) + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u))))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = dY_46_v * floor(h);
	t_1 = dY_46_u * floor(w);
	t_2 = dX_46_w * floor(d);
	t_3 = dY_46_w * floor(d);
	tmp = single(0.0);
	if (dX_46_u_m <= single(1900.0))
		tmp = log2(sqrt(max(((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v) + (t_2 * t_2)), ((t_3 * t_3) + ((t_0 * t_0) + (t_1 * t_1))))));
	else
		tmp = log2(sqrt(max(((t_2 ^ single(2.0)) + (((dX_46_u_m * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), (((floor(w) ^ single(2.0)) * dY_46_u) * dY_46_u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1900

   1. Initial program 71.5%

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

   3. Taylor expanded in dX.u around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 65.0

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

   5. Applied rewrites 65.0%

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

   if 1900 < dX.u

   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 dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.9

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in dY.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 57.7

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

   10. Applied rewrites 57.7%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 1900:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v + \left(dX.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dX.w \cdot \left\lfloor d\right\rfloor \right), \left(dY.w \cdot \left\lfloor d\right\rfloor \right) \cdot \left(dY.w \cdot \left\lfloor d\right\rfloor \right) + \left(\left(dY.v \cdot \left\lfloor h\right\rfloor \right) \cdot \left(dY.v \cdot \left\lfloor h\right\rfloor \right) + \left(dY.u \cdot \left\lfloor w\right\rfloor \right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u\_m \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\log_{2} \left({\left({\left(\mathsf{max}\left(t\_0, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left({\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0)))
   (if (<= dX.u_m 4.999999873689376e-5)
     (log2
      (pow
       (pow
        (fmax
         t_0
         (+
          (pow (* dY.w (floor d)) 2.0)
          (+ (pow (* dY.v (floor h)) 2.0) (pow (* dY.u (floor w)) 2.0))))
        0.25)
       2.0))
     (log2
      (sqrt
       (fmax
        (+ t_0 (+ (pow (* dX.u_m (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        (* (* (pow (floor w) 2.0) dY.u) dY.u)))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dX_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_u_m <= 4.999999873689376e-5f) {
		tmp = log2f(powf(powf(fmaxf(t_0, (powf((dY_46_w * floorf(d)), 2.0f) + (powf((dY_46_v * floorf(h)), 2.0f) + powf((dY_46_u * floorf(w)), 2.0f)))), 0.25f), 2.0f));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 + (powf((dX_46_u_m * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u_m <= Float32(4.999999873689376e-5))
		tmp = log2(((((t_0 != t_0) ? Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) : ((Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) != Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))) ? t_0 : max(t_0, Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + Float32((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))) ^ Float32(0.25)) ^ Float32(2.0)));
	else
		tmp = log2(sqrt(((Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))))) ? Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) : ((Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) != Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)) ? Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) : max(Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u))))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dX_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u_m <= single(4.999999873689376e-5))
		tmp = log2(((max(t_0, (((dY_46_w * floor(d)) ^ single(2.0)) + (((dY_46_v * floor(h)) ^ single(2.0)) + ((dY_46_u * floor(w)) ^ single(2.0))))) ^ single(0.25)) ^ single(2.0)));
	else
		tmp = log2(sqrt(max((t_0 + (((dX_46_u_m * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), (((floor(w) ^ single(2.0)) * dY_46_u) * dY_46_u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 4.99999987e-5

   1. Initial program 70.6%

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

   3. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.7

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

   10. Applied rewrites 56.7%

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

   12. Applied rewrites 56.7%

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

   if 4.99999987e-5 < dX.u

   1. Initial program 64.4%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in dY.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 57.8

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

   10. Applied rewrites 57.8%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\log_{2} \left({\left({\left(\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, {\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2} + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)\right)\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u\_m \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left({\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0)))
   (if (<= dX.u_m 4.999999873689376e-5)
     (log2
      (sqrt
       (fmax
        t_0
        (+
         (+ (pow (* dY.w (floor d)) 2.0) (pow (* dY.v (floor h)) 2.0))
         (pow (* dY.u (floor w)) 2.0)))))
     (log2
      (sqrt
       (fmax
        (+ t_0 (+ (pow (* dX.u_m (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        (* (* (pow (floor w) 2.0) dY.u) dY.u)))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dX_46_w * floorf(d)), 2.0f);
	float tmp;
	if (dX_46_u_m <= 4.999999873689376e-5f) {
		tmp = log2f(sqrtf(fmaxf(t_0, ((powf((dY_46_w * floorf(d)), 2.0f) + powf((dY_46_v * floorf(h)), 2.0f)) + powf((dY_46_u * floorf(w)), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 + (powf((dX_46_u_m * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), ((powf(floorf(w), 2.0f) * dY_46_u) * dY_46_u))));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w * floor(d)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u_m <= Float32(4.999999873689376e-5))
		tmp = log2(sqrt(((t_0 != t_0) ? Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? t_0 : max(t_0, Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))))) ? Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) : ((Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u) != Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u)) ? Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) : max(Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), Float32(Float32((floor(w) ^ Float32(2.0)) * dY_46_u) * dY_46_u))))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dX_46_w * floor(d)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u_m <= single(4.999999873689376e-5))
		tmp = log2(sqrt(max(t_0, ((((dY_46_w * floor(d)) ^ single(2.0)) + ((dY_46_v * floor(h)) ^ single(2.0))) + ((dY_46_u * floor(w)) ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((t_0 + (((dX_46_u_m * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), (((floor(w) ^ single(2.0)) * dY_46_u) * dY_46_u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 4.99999987e-5

   1. Initial program 70.6%

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

   3. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 55.8%

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

   8. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.7

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

   10. Applied rewrites 56.7%

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

   12. Applied rewrites 56.7%

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

   if 4.99999987e-5 < dX.u

   1. Initial program 64.4%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 58.8

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

   5. Applied rewrites 58.8%

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

   7. Applied rewrites 58.8%

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

   8. Taylor expanded in dY.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 57.8

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

   10. Applied rewrites 57.8%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), \left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dY.u\right) \cdot dY.u\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}\\ t_1 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u\_m \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + t\_1\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_0 + \left({\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), t\_1\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dX.w (floor d)) 2.0)) (t_1 (pow (* dY.v (floor h)) 2.0)))
   (if (<= dX.u_m 20.0)
     (log2
      (sqrt
       (fmax
        t_0
        (+
         (+ (pow (* dY.w (floor d)) 2.0) t_1)
         (pow (* dY.u (floor w)) 2.0)))))
     (log2
      (sqrt
       (fmax
        (+ t_0 (+ (pow (* dX.u_m (floor w)) 2.0) (pow (* dX.v (floor h)) 2.0)))
        t_1))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dX_46_w * floorf(d)), 2.0f);
	float t_1 = powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dX_46_u_m <= 20.0f) {
		tmp = log2f(sqrtf(fmaxf(t_0, ((powf((dY_46_w * floorf(d)), 2.0f) + t_1) + powf((dY_46_u * floorf(w)), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((t_0 + (powf((dX_46_u_m * floorf(w)), 2.0f) + powf((dX_46_v * floorf(h)), 2.0f))), t_1)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dX_46_w * floor(d)) ^ Float32(2.0)
	t_1 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u_m <= Float32(20.0))
		tmp = log2(sqrt(((t_0 != t_0) ? Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? t_0 : max(t_0, Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_1) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) != Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0))))) ? t_1 : ((t_1 != t_1) ? Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))) : max(Float32(t_0 + Float32((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) + (Float32(dX_46_v * floor(h)) ^ Float32(2.0)))), t_1)))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dX_46_w * floor(d)) ^ single(2.0);
	t_1 = (dY_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u_m <= single(20.0))
		tmp = log2(sqrt(max(t_0, ((((dY_46_w * floor(d)) ^ single(2.0)) + t_1) + ((dY_46_u * floor(w)) ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((t_0 + (((dX_46_u_m * floor(w)) ^ single(2.0)) + ((dX_46_v * floor(h)) ^ single(2.0)))), t_1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 20

   1. Initial program 70.7%

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

   3. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 56.3

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 56.3%

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

   8. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.3

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

   10. Applied rewrites 57.3%

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

   12. Applied rewrites 57.3%

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

   if 20 < dX.u

   1. Initial program 63.1%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 60.3

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

   5. Applied rewrites 60.3%

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

   7. Applied rewrites 60.3%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 20:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + \left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2} + {\left(dX.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right), {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 199999995904:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u\_m \cdot \left\lfloor w\right\rfloor \right)}^{2}, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + t\_0\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.v (floor h)) 2.0)))
   (if (<= dX.v 199999995904.0)
     (log2
      (sqrt
       (fmax
        (pow (* dX.u_m (floor w)) 2.0)
        (+
         (+ (pow (* dY.w (floor d)) 2.0) t_0)
         (pow (* dY.u (floor w)) 2.0)))))
     (log2 (sqrt (fmax (* (* (pow (floor h) 2.0) dX.v) dX.v) t_0))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dX_46_v <= 199999995904.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_u_m * floorf(w)), 2.0f), ((powf((dY_46_w * floorf(d)), 2.0f) + t_0) + powf((dY_46_u * floorf(w)), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), t_0)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(199999995904.0))
		tmp = log2(sqrt((((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) != (Float32(dX_46_u_m * floor(w)) ^ Float32(2.0))) ? Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? (Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)) : max((Float32(dX_46_u_m * floor(w)) ^ Float32(2.0)), Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) : max(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), t_0)))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dY_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(199999995904.0))
		tmp = log2(sqrt(max(((dX_46_u_m * floor(w)) ^ single(2.0)), ((((dY_46_w * floor(d)) ^ single(2.0)) + t_0) + ((dY_46_u * floor(w)) ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 199999996000

   1. Initial program 70.8%

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

   3. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 57.6

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

   5. Applied rewrites 57.6%

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

   7. Applied rewrites 57.6%

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

   if 199999996000 < dX.v

   1. Initial program 44.7%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 51.8

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

   10. Applied rewrites 51.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.v \leq 199999995904:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.u \cdot \left\lfloor w\right\rfloor \right)}^{2}, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.u\_m \leq 399999991808:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + t\_0\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\_m\right) \cdot dX.u\_m, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.v (floor h)) 2.0)))
   (if (<= dX.u_m 399999991808.0)
     (log2
      (sqrt
       (fmax
        (pow (* dX.w (floor d)) 2.0)
        (+
         (+ (pow (* dY.w (floor d)) 2.0) t_0)
         (pow (* dY.u (floor w)) 2.0)))))
     (log2 (sqrt (fmax (* (* (pow (floor w) 2.0) dX.u_m) dX.u_m) t_0))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dX_46_u_m <= 399999991808.0f) {
		tmp = log2f(sqrtf(fmaxf(powf((dX_46_w * floorf(d)), 2.0f), ((powf((dY_46_w * floorf(d)), 2.0f) + t_0) + powf((dY_46_u * floorf(w)), 2.0f)))));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(w), 2.0f) * dX_46_u_m) * dX_46_u_m), t_0)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_u_m <= Float32(399999991808.0))
		tmp = log2(sqrt((((Float32(dX_46_w * floor(d)) ^ Float32(2.0)) != (Float32(dX_46_w * floor(d)) ^ Float32(2.0))) ? Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) : ((Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))) != Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0)))) ? (Float32(dX_46_w * floor(d)) ^ Float32(2.0)) : max((Float32(dX_46_w * floor(d)) ^ Float32(2.0)), Float32(Float32((Float32(dY_46_w * floor(d)) ^ Float32(2.0)) + t_0) + (Float32(dY_46_u * floor(w)) ^ Float32(2.0))))))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m) != Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m) : max(Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m), t_0)))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dY_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_u_m <= single(399999991808.0))
		tmp = log2(sqrt(max(((dX_46_w * floor(d)) ^ single(2.0)), ((((dY_46_w * floor(d)) ^ single(2.0)) + t_0) + ((dY_46_u * floor(w)) ^ single(2.0))))));
	else
		tmp = log2(sqrt(max((((floor(w) ^ single(2.0)) * dX_46_u_m) * dX_46_u_m), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 399999992000

   1. Initial program 70.5%

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

   3. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-floor.f32 55.0

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

   5. Applied rewrites 55.0%

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 57.1

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

   10. Applied rewrites 57.1%

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

   12. Applied rewrites 57.1%

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

   if 399999992000 < dX.u

   1. Initial program 56.0%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 55.3

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

   10. Applied rewrites 55.3%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;dX.u \leq 399999991808:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left({\left(dX.w \cdot \left\lfloor d\right\rfloor \right)}^{2}, \left({\left(dY.w \cdot \left\lfloor d\right\rfloor \right)}^{2} + {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right) + {\left(dY.u \cdot \left\lfloor w\right\rfloor \right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\right) \cdot dX.u, {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.w \leq 250:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor w\right\rfloor \right)}^{2} \cdot dX.u\_m\right) \cdot dX.u\_m, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.v (floor h)) 2.0)))
   (if (<= dX.w 250.0)
     (log2 (sqrt (fmax (* (* (pow (floor w) 2.0) dX.u_m) dX.u_m) t_0)))
     (log2 (sqrt (fmax (* (* (pow (floor d) 2.0) dX.w) dX.w) t_0))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dX_46_w <= 250.0f) {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(w), 2.0f) * dX_46_u_m) * dX_46_u_m), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(d), 2.0f) * dX_46_w) * dX_46_w), t_0)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_w <= Float32(250.0))
		tmp = log2(sqrt(((Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m) != Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m) : max(Float32(Float32((floor(w) ^ Float32(2.0)) * dX_46_u_m) * dX_46_u_m), t_0)))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) != Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) : max(Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w), t_0)))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dY_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_w <= single(250.0))
		tmp = log2(sqrt(max((((floor(w) ^ single(2.0)) * dX_46_u_m) * dX_46_u_m), t_0)));
	else
		tmp = log2(sqrt(max((((floor(d) ^ single(2.0)) * dX_46_w) * dX_46_w), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.w < 250

   1. Initial program 67.6%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 55.5

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.5%

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

   8. Taylor expanded in dX.u around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 39.5

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

   10. Applied rewrites 39.5%

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

   if 250 < dX.w

   1. Initial program 72.1%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 58.2

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

   5. Applied rewrites 58.2%

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 49.0

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

   10. Applied rewrites 49.0%

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

Alternative 10: 39.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} dX.u_m = \left|dX.u\right| \\ \begin{array}{l} t_0 := {\left(dY.v \cdot \left\lfloor h\right\rfloor \right)}^{2}\\ \mathbf{if}\;dX.v \leq 30000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor d\right\rfloor \right)}^{2} \cdot dX.w\right) \cdot dX.w, t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left({\left(\left\lfloor h\right\rfloor \right)}^{2} \cdot dX.v\right) \cdot dX.v, t\_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (let* ((t_0 (pow (* dY.v (floor h)) 2.0)))
   (if (<= dX.v 30000.0)
     (log2 (sqrt (fmax (* (* (pow (floor d) 2.0) dX.w) dX.w) t_0)))
     (log2 (sqrt (fmax (* (* (pow (floor h) 2.0) dX.v) dX.v) t_0))))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = powf((dY_46_v * floorf(h)), 2.0f);
	float tmp;
	if (dX_46_v <= 30000.0f) {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(d), 2.0f) * dX_46_w) * dX_46_w), t_0)));
	} else {
		tmp = log2f(sqrtf(fmaxf(((powf(floorf(h), 2.0f) * dX_46_v) * dX_46_v), t_0)));
	}
	return tmp;
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(dY_46_v * floor(h)) ^ Float32(2.0)
	tmp = Float32(0.0)
	if (dX_46_v <= Float32(30000.0))
		tmp = log2(sqrt(((Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) != Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) : max(Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w), t_0)))));
	else
		tmp = log2(sqrt(((Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) != Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v)) ? t_0 : ((t_0 != t_0) ? Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v) : max(Float32(Float32((floor(h) ^ Float32(2.0)) * dX_46_v) * dX_46_v), t_0)))));
	end
	return tmp
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp_2 = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = (dY_46_v * floor(h)) ^ single(2.0);
	tmp = single(0.0);
	if (dX_46_v <= single(30000.0))
		tmp = log2(sqrt(max((((floor(d) ^ single(2.0)) * dX_46_w) * dX_46_w), t_0)));
	else
		tmp = log2(sqrt(max((((floor(h) ^ single(2.0)) * dX_46_v) * dX_46_v), t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if dX.v < 3e4

   1. Initial program 70.9%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 56.5

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

   5. Applied rewrites 56.5%

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

   7. Applied rewrites 56.5%

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

   8. Taylor expanded in dX.w around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 36.9

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

   10. Applied rewrites 36.9%

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

   if 3e4 < dX.v

   1. Initial program 58.2%

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

   3. Taylor expanded in dY.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in dX.v around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-floor.f32 43.5

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

   10. Applied rewrites 43.5%

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

Alternative 11: 36.0% accurate, 2.4× speedup

Math

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

FPCore

dX.u_m = (fabs.f32 dX.u)
(FPCore (w h d dX.u_m dX.v dX.w dY.u dY.v dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax (* (* (pow (floor d) 2.0) dX.w) dX.w) (pow (* dY.v (floor h)) 2.0)))))

C

dX.u_m = fabs(dX_46_u);
float code(float w, float h, float d, float dX_46_u_m, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	return log2f(sqrtf(fmaxf(((powf(floorf(d), 2.0f) * dX_46_w) * dX_46_w), powf((dY_46_v * floorf(h)), 2.0f))));
}

Julia

dX.u_m = abs(dX_46_u)
function code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	return log2(sqrt(((Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) != Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w)) ? (Float32(dY_46_v * floor(h)) ^ Float32(2.0)) : (((Float32(dY_46_v * floor(h)) ^ Float32(2.0)) != (Float32(dY_46_v * floor(h)) ^ Float32(2.0))) ? Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w) : max(Float32(Float32((floor(d) ^ Float32(2.0)) * dX_46_w) * dX_46_w), (Float32(dY_46_v * floor(h)) ^ Float32(2.0)))))))
end

MATLAB

dX.u_m = abs(dX_46_u);
function tmp = code(w, h, d, dX_46_u_m, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	tmp = log2(sqrt(max((((floor(d) ^ single(2.0)) * dX_46_w) * dX_46_w), ((dY_46_v * floor(h)) ^ single(2.0)))));
end

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in dY.v around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-floor.f32 56.1

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

5. Applied rewrites 56.1%

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

7. Applied rewrites 56.1%

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

8. Taylor expanded in dX.w around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower-floor.f32 35.8

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

10. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2024268 
(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)))))))