Isotropic LOD (LOD)

Percentage Accurate: 67.6% → 78.1%

Time: 14.5s

Alternatives: 16

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 67.6% 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(fmax(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + Float32(t_1 * t_1)) + Float32(t_3 * t_3)))))
end

MATLAB

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

Alternative 1: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_5 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ t_6 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_7 := \log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_3 \cdot t\_3\right) + t\_6 \cdot t\_6, \left(t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4\right)}\right)\\ \mathbf{if}\;t\_7 \leq 100:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_5 \cdot t\_5\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v_m))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor d) dY.w))
        (t_5 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0))))
        (t_6 (* (floor d) dX.w))
        (t_7
         (log2
          (sqrt
           (fmax
            (+ (+ (* t_0 t_0) (* t_3 t_3)) (* t_6 t_6))
            (+ (+ (* t_1 t_1) (* t_2 t_2)) (* t_4 t_4)))))))
   (if (<= t_7 100.0) t_7 (log2 (* t_5 t_5)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v_m;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float t_6 = floorf(d) * dX_46_w;
	float t_7 = log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_3 * t_3)) + (t_6 * t_6)), (((t_1 * t_1) + (t_2 * t_2)) + (t_4 * t_4)))));
	float tmp;
	if (t_7 <= 100.0f) {
		tmp = t_7;
	} else {
		tmp = log2f((t_5 * t_5));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v_m)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	t_6 = Float32(floor(d) * dX_46_w)
	t_7 = log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3)) + Float32(t_6 * t_6)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)))))
	tmp = Float32(0.0)
	if (t_7 <= Float32(100.0))
		tmp = t_7;
	else
		tmp = log2(Float32(t_5 * t_5));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 67.6%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 2: 69.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ t_3 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_6 := t\_5 \cdot t\_5\\ t_7 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_8 := \left(t\_0 \cdot t\_0 + t\_4 \cdot t\_4\right) + t\_7 \cdot t\_7\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_8, \left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right) + t\_6\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_8, \mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_3\right) \cdot \left(t\_3 - t\_1\right) + t\_6\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_2 \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0))))
        (t_3 (* (floor h) dY.v_m))
        (t_4 (* (floor h) dX.v))
        (t_5 (* (floor d) dY.w))
        (t_6 (* t_5 t_5))
        (t_7 (* (floor d) dX.w))
        (t_8 (+ (+ (* t_0 t_0) (* t_4 t_4)) (* t_7 t_7))))
   (if (<= (log2 (sqrt (fmax t_8 (+ (+ (* t_1 t_1) (* t_3 t_3)) t_6)))) 100.0)
     (log2 (sqrt (fmax t_8 (+ (* (fma (floor w) dY.u t_3) (- t_3 t_1)) t_6))))
     (log2 (* t_2 t_2)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float t_3 = floorf(h) * dY_46_v_m;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w;
	float t_6 = t_5 * t_5;
	float t_7 = floorf(d) * dX_46_w;
	float t_8 = ((t_0 * t_0) + (t_4 * t_4)) + (t_7 * t_7);
	float tmp;
	if (log2f(sqrtf(fmaxf(t_8, (((t_1 * t_1) + (t_3 * t_3)) + t_6)))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(t_8, ((fmaf(floorf(w), dY_46_u, t_3) * (t_3 - t_1)) + t_6))));
	} else {
		tmp = log2f((t_2 * t_2));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	t_3 = Float32(floor(h) * dY_46_v_m)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w)
	t_6 = Float32(t_5 * t_5)
	t_7 = Float32(floor(d) * dX_46_w)
	t_8 = Float32(Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)) + Float32(t_7 * t_7))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(t_8, Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + t_6)))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(t_8, Float32(Float32(fma(floor(w), dY_46_u, t_3) * Float32(t_3 - t_1)) + t_6))));
	else
		tmp = log2(Float32(t_2 * t_2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 67.6%

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

   3. Applied rewrites 59.6%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 3: 69.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ t_3 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_6 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_7 := t\_6 \cdot t\_6\\ t_8 := \left(t\_1 \cdot t\_1 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_4 \cdot t\_4\right) + t\_7, t\_8\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, t\_4\right) \cdot \left(t\_0 - t\_4\right) + t\_7, t\_8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_2 \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0))))
        (t_3 (* (floor h) dY.v_m))
        (t_4 (* (floor h) dX.v))
        (t_5 (* (floor d) dY.w))
        (t_6 (* (floor d) dX.w))
        (t_7 (* t_6 t_6))
        (t_8 (+ (+ (* t_1 t_1) (* t_3 t_3)) (* t_5 t_5))))
   (if (<= (log2 (sqrt (fmax (+ (+ (* t_0 t_0) (* t_4 t_4)) t_7) t_8))) 100.0)
     (log2 (sqrt (fmax (+ (* (fma (floor w) dX.u t_4) (- t_0 t_4)) t_7) t_8)))
     (log2 (* t_2 t_2)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float t_3 = floorf(h) * dY_46_v_m;
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w;
	float t_6 = floorf(d) * dX_46_w;
	float t_7 = t_6 * t_6;
	float t_8 = ((t_1 * t_1) + (t_3 * t_3)) + (t_5 * t_5);
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_4 * t_4)) + t_7), t_8))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(((fmaf(floorf(w), dX_46_u, t_4) * (t_0 - t_4)) + t_7), t_8)));
	} else {
		tmp = log2f((t_2 * t_2));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	t_3 = Float32(floor(h) * dY_46_v_m)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w)
	t_6 = Float32(floor(d) * dX_46_w)
	t_7 = Float32(t_6 * t_6)
	t_8 = Float32(Float32(Float32(t_1 * t_1) + Float32(t_3 * t_3)) + Float32(t_5 * t_5))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)) + t_7), t_8))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(Float32(Float32(fma(floor(w), dX_46_u, t_4) * Float32(t_0 - t_4)) + t_7), t_8)));
	else
		tmp = log2(Float32(t_2 * t_2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 4: 68.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_3 := \mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_2\right)\\ t_4 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_5 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_6 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_7 := t\_6 \cdot t\_6\\ t_8 := t\_5 \cdot t\_5\\ t_9 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_4 \cdot t\_4\right) + t\_7, \left(t\_1 \cdot t\_1 + t\_2 \cdot t\_2\right) + t\_8\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, t\_4\right) \cdot \left(t\_0 - t\_4\right) + t\_7, t\_3 \cdot t\_3 + t\_8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_9 \cdot t\_9\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v_m))
        (t_3 (fma (floor w) dY.u t_2))
        (t_4 (* (floor h) dX.v))
        (t_5 (* (floor d) dY.w))
        (t_6 (* (floor d) dX.w))
        (t_7 (* t_6 t_6))
        (t_8 (* t_5 t_5))
        (t_9 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0)))))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_0 t_0) (* t_4 t_4)) t_7)
           (+ (+ (* t_1 t_1) (* t_2 t_2)) t_8))))
        100.0)
     (log2
      (sqrt
       (fmax
        (+ (* (fma (floor w) dX.u t_4) (- t_0 t_4)) t_7)
        (+ (* t_3 t_3) t_8))))
     (log2 (* t_9 t_9)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v_m;
	float t_3 = fmaf(floorf(w), dY_46_u, t_2);
	float t_4 = floorf(h) * dX_46_v;
	float t_5 = floorf(d) * dY_46_w;
	float t_6 = floorf(d) * dX_46_w;
	float t_7 = t_6 * t_6;
	float t_8 = t_5 * t_5;
	float t_9 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_4 * t_4)) + t_7), (((t_1 * t_1) + (t_2 * t_2)) + t_8)))) <= 100.0f) {
		tmp = log2f(sqrtf(fmaxf(((fmaf(floorf(w), dX_46_u, t_4) * (t_0 - t_4)) + t_7), ((t_3 * t_3) + t_8))));
	} else {
		tmp = log2f((t_9 * t_9));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v_m)
	t_3 = fma(floor(w), dY_46_u, t_2)
	t_4 = Float32(floor(h) * dX_46_v)
	t_5 = Float32(floor(d) * dY_46_w)
	t_6 = Float32(floor(d) * dX_46_w)
	t_7 = Float32(t_6 * t_6)
	t_8 = Float32(t_5 * t_5)
	t_9 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_4 * t_4)) + t_7), Float32(Float32(Float32(t_1 * t_1) + Float32(t_2 * t_2)) + t_8)))) <= Float32(100.0))
		tmp = log2(sqrt(fmax(Float32(Float32(fma(floor(w), dX_46_u, t_4) * Float32(t_0 - t_4)) + t_7), Float32(Float32(t_3 * t_3) + t_8))));
	else
		tmp = log2(Float32(t_9 * t_9));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 59.0%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 5: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_6 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_7 := t\_6 \cdot t\_6\\ t_8 := \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 1\right)\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_7\right) + t\_4 \cdot t\_4, \left(t\_1 \cdot t\_1 + t\_5 \cdot t\_5\right) + t\_2 \cdot t\_2\right)}\right) \leq 63.95000076293945:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|t\_0\right| - t\_4}{1} \cdot 1, t\_0 - t\_4, t\_7\right), t\_8 \cdot t\_8\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_3 \cdot t\_3\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor d) dY.w))
        (t_3 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0))))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor h) dY.v_m))
        (t_6 (* (floor h) dX.v))
        (t_7 (* t_6 t_6))
        (t_8 (fma (floor h) dY.v_m 1.0)))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_0 t_0) t_7) (* t_4 t_4))
           (+ (+ (* t_1 t_1) (* t_5 t_5)) (* t_2 t_2)))))
        63.95000076293945)
     (log2
      (sqrt
       (fmax
        (fma (* (/ (- (fabs t_0) t_4) 1.0) 1.0) (- t_0 t_4) t_7)
        (* t_8 t_8))))
     (log2 (* t_3 t_3)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(h) * dY_46_v_m;
	float t_6 = floorf(h) * dX_46_v;
	float t_7 = t_6 * t_6;
	float t_8 = fmaf(floorf(h), dY_46_v_m, 1.0f);
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + t_7) + (t_4 * t_4)), (((t_1 * t_1) + (t_5 * t_5)) + (t_2 * t_2))))) <= 63.95000076293945f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((((fabsf(t_0) - t_4) / 1.0f) * 1.0f), (t_0 - t_4), t_7), (t_8 * t_8))));
	} else {
		tmp = log2f((t_3 * t_3));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	t_4 = Float32(floor(d) * dX_46_w)
	t_5 = Float32(floor(h) * dY_46_v_m)
	t_6 = Float32(floor(h) * dX_46_v)
	t_7 = Float32(t_6 * t_6)
	t_8 = fma(floor(h), dY_46_v_m, Float32(1.0))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + t_7) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_1 * t_1) + Float32(t_5 * t_5)) + Float32(t_2 * t_2))))) <= Float32(63.95000076293945))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(Float32(abs(t_0) - t_4) / Float32(1.0)) * Float32(1.0)), Float32(t_0 - t_4), t_7), Float32(t_8 * t_8))));
	else
		tmp = log2(Float32(t_3 * t_3));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 63.9500008

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   5. Applied rewrites 58.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \frac{\mathsf{fma}\left(\left\lfloor d\right\rfloor , dX.w, \left|\left\lfloor w\right\rfloor \cdot dX.u\right|\right)}{\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)}}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   6. Taylor expanded in dX.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \color{blue}{1}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   8. Applied rewrites 53.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \color{blue}{1}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   10. Applied rewrites 47.2%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot 1, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \color{blue}{\mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v, 1\right) \cdot \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v, 1\right)}\right)}\right) \]

   if 63.9500008 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 6: 52.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_1 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_2 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ \mathbf{if}\;dY.v\_m \leq 950000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|t\_1\right| - t\_3}{1} \cdot 1, t\_1 - t\_3, t\_0 \cdot t\_0\right), \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 1\right) \cdot 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_2 \cdot t\_2\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dX.v))
        (t_1 (* (floor w) dX.u))
        (t_2 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0))))
        (t_3 (* (floor d) dX.w)))
   (if (<= dY.v_m 950000000.0)
     (log2
      (sqrt
       (fmax
        (fma (* (/ (- (fabs t_1) t_3) 1.0) 1.0) (- t_1 t_3) (* t_0 t_0))
        (* (fma (floor h) dY.v_m 1.0) 1.0))))
     (log2 (* t_2 t_2)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(h) * dX_46_v;
	float t_1 = floorf(w) * dX_46_u;
	float t_2 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float t_3 = floorf(d) * dX_46_w;
	float tmp;
	if (dY_46_v_m <= 950000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((((fabsf(t_1) - t_3) / 1.0f) * 1.0f), (t_1 - t_3), (t_0 * t_0)), (fmaf(floorf(h), dY_46_v_m, 1.0f) * 1.0f))));
	} else {
		tmp = log2f((t_2 * t_2));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(h) * dX_46_v)
	t_1 = Float32(floor(w) * dX_46_u)
	t_2 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	t_3 = Float32(floor(d) * dX_46_w)
	tmp = Float32(0.0)
	if (dY_46_v_m <= Float32(950000000.0))
		tmp = log2(sqrt(fmax(fma(Float32(Float32(Float32(abs(t_1) - t_3) / Float32(1.0)) * Float32(1.0)), Float32(t_1 - t_3), Float32(t_0 * t_0)), Float32(fma(floor(h), dY_46_v_m, Float32(1.0)) * Float32(1.0)))));
	else
		tmp = log2(Float32(t_2 * t_2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 9.5e8

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   5. Applied rewrites 58.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \frac{\mathsf{fma}\left(\left\lfloor d\right\rfloor , dX.w, \left|\left\lfloor w\right\rfloor \cdot dX.u\right|\right)}{\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)}}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   6. Taylor expanded in dX.w around inf

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \color{blue}{1}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   8. Applied rewrites 53.9%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \color{blue}{1}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   10. Applied rewrites 41.1%

       \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot 1, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \color{blue}{\mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v, 1\right) \cdot 1}\right)}\right) \]

   if 9.5e8 < dY.v

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 7: 50.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_2 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_3 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_4 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_5 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_6 := \mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_3\right)\\ t_7 := t\_1 \cdot t\_1\\ t_8 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_5 \cdot t\_5 + t\_2 \cdot t\_2\right) + t\_4 \cdot t\_4, \left(t\_0 \cdot t\_0 + t\_7\right) + t\_3 \cdot t\_3\right)}\right) \leq 100:\\ \;\;\;\;\log_{2} \left(\sqrt{e^{\log \left(\mathsf{max}\left(dX.u, \mathsf{fma}\left(t\_6, t\_6, t\_7\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_8 \cdot t\_8\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dY.u))
        (t_1 (* (floor h) dY.v_m))
        (t_2 (* (floor h) dX.v))
        (t_3 (* (floor d) dY.w))
        (t_4 (* (floor d) dX.w))
        (t_5 (* (floor w) dX.u))
        (t_6 (fma (floor w) dY.u t_3))
        (t_7 (* t_1 t_1))
        (t_8 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0)))))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_5 t_5) (* t_2 t_2)) (* t_4 t_4))
           (+ (+ (* t_0 t_0) t_7) (* t_3 t_3)))))
        100.0)
     (log2 (sqrt (exp (log (fmax dX.u (fma t_6 t_6 t_7))))))
     (log2 (* t_8 t_8)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dY_46_u;
	float t_1 = floorf(h) * dY_46_v_m;
	float t_2 = floorf(h) * dX_46_v;
	float t_3 = floorf(d) * dY_46_w;
	float t_4 = floorf(d) * dX_46_w;
	float t_5 = floorf(w) * dX_46_u;
	float t_6 = fmaf(floorf(w), dY_46_u, t_3);
	float t_7 = t_1 * t_1;
	float t_8 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_5 * t_5) + (t_2 * t_2)) + (t_4 * t_4)), (((t_0 * t_0) + t_7) + (t_3 * t_3))))) <= 100.0f) {
		tmp = log2f(sqrtf(expf(logf(fmaxf(dX_46_u, fmaf(t_6, t_6, t_7))))));
	} else {
		tmp = log2f((t_8 * t_8));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dY_46_v_m)
	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)
	t_6 = fma(floor(w), dY_46_u, t_3)
	t_7 = Float32(t_1 * t_1)
	t_8 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_5 * t_5) + Float32(t_2 * t_2)) + Float32(t_4 * t_4)), Float32(Float32(Float32(t_0 * t_0) + t_7) + Float32(t_3 * t_3))))) <= Float32(100.0))
		tmp = log2(sqrt(exp(log(fmax(dX_46_u, fma(t_6, t_6, t_7))))));
	else
		tmp = log2(Float32(t_8 * t_8));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 100

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 45.5%

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

   if 100 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 8: 44.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_1 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := dY.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dX.u \leq 5000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_2\right), \left\lfloor w\right\rfloor \cdot dY.u - t\_2, t\_0 \cdot t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \left\lfloor w\right\rfloor , t\_1\right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor - t\_1\right), \mathsf{fma}\left(dY.u, \left\lfloor w\right\rfloor , t\_3\right) \cdot \left(dY.u \cdot \left\lfloor w\right\rfloor - t\_3\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v_m))
        (t_1 (* dX.w (floor d)))
        (t_2 (* (floor d) dY.w))
        (t_3 (* dY.w (floor d))))
   (if (<= dX.u 5000000.0)
     (log2
      (sqrt
       (fmax
        (fma (floor w) dX.u (* (floor h) dX.v))
        (fma
         (fma (floor w) dY.u t_2)
         (- (* (floor w) dY.u) t_2)
         (* t_0 t_0)))))
     (log2
      (sqrt
       (fmax
        (* (fma dX.u (floor w) t_1) (- (* dX.u (floor w)) t_1))
        (* (fma dY.u (floor w) t_3) (- (* dY.u (floor w)) t_3))))))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v_m;
	float t_1 = dX_46_w * floorf(d);
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = dY_46_w * floorf(d);
	float tmp;
	if (dX_46_u <= 5000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(floorf(w), dX_46_u, (floorf(h) * dX_46_v)), fmaf(fmaf(floorf(w), dY_46_u, t_2), ((floorf(w) * dY_46_u) - t_2), (t_0 * t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((fmaf(dX_46_u, floorf(w), t_1) * ((dX_46_u * floorf(w)) - t_1)), (fmaf(dY_46_u, floorf(w), t_3) * ((dY_46_u * floorf(w)) - t_3)))));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v_m)
	t_1 = Float32(dX_46_w * floor(d))
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(dY_46_w * floor(d))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(5000000.0))
		tmp = log2(sqrt(fmax(fma(floor(w), dX_46_u, Float32(floor(h) * dX_46_v)), fma(fma(floor(w), dY_46_u, t_2), Float32(Float32(floor(w) * dY_46_u) - t_2), Float32(t_0 * t_0)))));
	else
		tmp = log2(sqrt(fmax(Float32(fma(dX_46_u, floor(w), t_1) * Float32(Float32(dX_46_u * floor(w)) - t_1)), Float32(fma(dY_46_u, floor(w), t_3) * Float32(Float32(dY_46_u * floor(w)) - t_3)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 5e6

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   5. Applied rewrites 58.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \frac{\mathsf{fma}\left(\left\lfloor d\right\rfloor , dX.w, \left|\left\lfloor w\right\rfloor \cdot dX.u\right|\right)}{\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)}}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   6. Taylor expanded in dX.w around 0

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 34.9%

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

   if 5e6 < dX.u

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   4. Taylor expanded in dX.v around 0

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

   6. Applied rewrites 41.4%

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

   7. Taylor expanded in dY.v around 0

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

   9. Applied rewrites 31.3%

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

Alternative 9: 39.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_1 := dX.w \cdot \left\lfloor d\right\rfloor \\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ \mathbf{if}\;dX.u \leq 10000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_2\right), \left\lfloor w\right\rfloor \cdot dY.u - t\_2, t\_0 \cdot t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \left\lfloor w\right\rfloor , t\_1\right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor - t\_1\right), 2 \cdot \left({dY.u}^{2} \cdot \left(dY.w \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v_m))
        (t_1 (* dX.w (floor d)))
        (t_2 (* (floor d) dY.w)))
   (if (<= dX.u 10000000000.0)
     (log2
      (sqrt
       (fmax
        (fma (floor w) dX.u (* (floor h) dX.v))
        (fma
         (fma (floor w) dY.u t_2)
         (- (* (floor w) dY.u) t_2)
         (* t_0 t_0)))))
     (log2
      (sqrt
       (fmax
        (* (fma dX.u (floor w) t_1) (- (* dX.u (floor w)) t_1))
        (* 2.0 (* (pow dY.u 2.0) (* dY.w (floor w))))))))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v_m;
	float t_1 = dX_46_w * floorf(d);
	float t_2 = floorf(d) * dY_46_w;
	float tmp;
	if (dX_46_u <= 10000000000.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(floorf(w), dX_46_u, (floorf(h) * dX_46_v)), fmaf(fmaf(floorf(w), dY_46_u, t_2), ((floorf(w) * dY_46_u) - t_2), (t_0 * t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((fmaf(dX_46_u, floorf(w), t_1) * ((dX_46_u * floorf(w)) - t_1)), (2.0f * (powf(dY_46_u, 2.0f) * (dY_46_w * floorf(w)))))));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v_m)
	t_1 = Float32(dX_46_w * floor(d))
	t_2 = Float32(floor(d) * dY_46_w)
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(10000000000.0))
		tmp = log2(sqrt(fmax(fma(floor(w), dX_46_u, Float32(floor(h) * dX_46_v)), fma(fma(floor(w), dY_46_u, t_2), Float32(Float32(floor(w) * dY_46_u) - t_2), Float32(t_0 * t_0)))));
	else
		tmp = log2(sqrt(fmax(Float32(fma(dX_46_u, floor(w), t_1) * Float32(Float32(dX_46_u * floor(w)) - t_1)), Float32(Float32(2.0) * Float32((dY_46_u ^ Float32(2.0)) * Float32(dY_46_w * floor(w)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1e10

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   5. Applied rewrites 58.7%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \frac{\mathsf{fma}\left(\left\lfloor d\right\rfloor , dX.w, \left|\left\lfloor w\right\rfloor \cdot dX.u\right|\right)}{\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)}}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

   6. Taylor expanded in dX.w around 0

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

   8. Applied rewrites 52.2%

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

   10. Applied rewrites 34.9%

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

   if 1e10 < dX.u

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   4. Taylor expanded in dX.v around 0

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

   6. Applied rewrites 41.4%

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

   8. Applied rewrites 31.8%

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

   9. Taylor expanded in dY.u around inf

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

   11. Applied rewrites 22.2%

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

Alternative 10: 36.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_2 := dX.w \cdot \left\lfloor d\right\rfloor \\ \mathbf{if}\;dX.u \leq 10000000000:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_1\right), \left\lfloor w\right\rfloor \cdot dY.u - t\_1, t\_0 \cdot t\_0\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u, \left\lfloor w\right\rfloor , t\_2\right) \cdot \left(dX.u \cdot \left\lfloor w\right\rfloor - t\_2\right), 2 \cdot \left({dY.u}^{2} \cdot \left(dY.w \cdot \left\lfloor w\right\rfloor \right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor h) dY.v_m))
        (t_1 (* (floor d) dY.w))
        (t_2 (* dX.w (floor d))))
   (if (<= dX.u 10000000000.0)
     (log2
      (sqrt
       (fmax
        2.0
        (fma
         (fma (floor w) dY.u t_1)
         (- (* (floor w) dY.u) t_1)
         (* t_0 t_0)))))
     (log2
      (sqrt
       (fmax
        (* (fma dX.u (floor w) t_2) (- (* dX.u (floor w)) t_2))
        (* 2.0 (* (pow dY.u 2.0) (* dY.w (floor w))))))))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(h) * dY_46_v_m;
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = dX_46_w * floorf(d);
	float tmp;
	if (dX_46_u <= 10000000000.0f) {
		tmp = log2f(sqrtf(fmaxf(2.0f, fmaf(fmaf(floorf(w), dY_46_u, t_1), ((floorf(w) * dY_46_u) - t_1), (t_0 * t_0)))));
	} else {
		tmp = log2f(sqrtf(fmaxf((fmaf(dX_46_u, floorf(w), t_2) * ((dX_46_u * floorf(w)) - t_2)), (2.0f * (powf(dY_46_u, 2.0f) * (dY_46_w * floorf(w)))))));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(h) * dY_46_v_m)
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(dX_46_w * floor(d))
	tmp = Float32(0.0)
	if (dX_46_u <= Float32(10000000000.0))
		tmp = log2(sqrt(fmax(Float32(2.0), fma(fma(floor(w), dY_46_u, t_1), Float32(Float32(floor(w) * dY_46_u) - t_1), Float32(t_0 * t_0)))));
	else
		tmp = log2(sqrt(fmax(Float32(fma(dX_46_u, floor(w), t_2) * Float32(Float32(dX_46_u * floor(w)) - t_2)), Float32(Float32(2.0) * Float32((dY_46_u ^ Float32(2.0)) * Float32(dY_46_w * floor(w)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1e10

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   4. Taylor expanded in dX.v around 0

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

   6. Applied rewrites 41.4%

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

   8. Applied rewrites 37.6%

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

   if 1e10 < dX.u

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   4. Taylor expanded in dX.v around 0

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

   6. Applied rewrites 41.4%

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

   8. Applied rewrites 31.8%

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

   9. Taylor expanded in dY.u around inf

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

   11. Applied rewrites 22.2%

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

Alternative 11: 36.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_2 := \left\lfloor h\right\rfloor \cdot dY.v\_m\\ t_3 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_4 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := t\_2 \cdot t\_2\\ t_7 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ \mathbf{if}\;\log_{2} \left(\sqrt{\mathsf{max}\left(\left(t\_0 \cdot t\_0 + t\_3 \cdot t\_3\right) + t\_5 \cdot t\_5, \left(t\_1 \cdot t\_1 + t\_6\right) + t\_4 \cdot t\_4\right)}\right) \leq 63.220001220703125:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, t\_4\right), t\_1 - t\_4, t\_6\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(t\_7 \cdot t\_7\right)\\ \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (* (floor w) dX.u))
        (t_1 (* (floor w) dY.u))
        (t_2 (* (floor h) dY.v_m))
        (t_3 (* (floor h) dX.v))
        (t_4 (* (floor d) dY.w))
        (t_5 (* (floor d) dX.w))
        (t_6 (* t_2 t_2))
        (t_7 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0)))))
   (if (<=
        (log2
         (sqrt
          (fmax
           (+ (+ (* t_0 t_0) (* t_3 t_3)) (* t_5 t_5))
           (+ (+ (* t_1 t_1) t_6) (* t_4 t_4)))))
        63.220001220703125)
     (log2 (sqrt (fmax 2.0 (fma (fma (floor w) dY.u t_4) (- t_1 t_4) t_6))))
     (log2 (* t_7 t_7)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = floorf(w) * dX_46_u;
	float t_1 = floorf(w) * dY_46_u;
	float t_2 = floorf(h) * dY_46_v_m;
	float t_3 = floorf(h) * dX_46_v;
	float t_4 = floorf(d) * dY_46_w;
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = t_2 * t_2;
	float t_7 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	float tmp;
	if (log2f(sqrtf(fmaxf((((t_0 * t_0) + (t_3 * t_3)) + (t_5 * t_5)), (((t_1 * t_1) + t_6) + (t_4 * t_4))))) <= 63.220001220703125f) {
		tmp = log2f(sqrtf(fmaxf(2.0f, fmaf(fmaf(floorf(w), dY_46_u, t_4), (t_1 - t_4), t_6))));
	} else {
		tmp = log2f((t_7 * t_7));
	}
	return tmp;
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * dY_46_u)
	t_2 = Float32(floor(h) * dY_46_v_m)
	t_3 = Float32(floor(h) * dX_46_v)
	t_4 = Float32(floor(d) * dY_46_w)
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(t_2 * t_2)
	t_7 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	tmp = Float32(0.0)
	if (log2(sqrt(fmax(Float32(Float32(Float32(t_0 * t_0) + Float32(t_3 * t_3)) + Float32(t_5 * t_5)), Float32(Float32(Float32(t_1 * t_1) + t_6) + Float32(t_4 * t_4))))) <= Float32(63.220001220703125))
		tmp = log2(sqrt(fmax(Float32(2.0), fma(fma(floor(w), dY_46_u, t_4), Float32(t_1 - t_4), t_6))));
	else
		tmp = log2(Float32(t_7 * t_7));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))))) < 63.2200012

   1. Initial program 67.6%

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

   3. Applied rewrites 49.9%

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

   4. Taylor expanded in dX.v around 0

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

   6. Applied rewrites 41.4%

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

   8. Applied rewrites 37.6%

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

   if 63.2200012 < (log2.f32 (sqrt.f32 (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w))))))

   1. Initial program 67.6%

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

   3. Applied rewrites 60.2%

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

   5. Applied rewrites 64.6%

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

   7. Applied rewrites 36.0%

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

Alternative 12: 36.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\\ \log_{2} \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (let* ((t_0 (sqrt (fmax 2.0 (fma (floor h) dY.v_m 2.0)))))
   (log2 (* t_0 t_0))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	float t_0 = sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f)));
	return log2f((t_0 * t_0));
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	t_0 = sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0))))
	return log2(Float32(t_0 * t_0))
end

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 60.2%

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

5. Applied rewrites 64.6%

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

7. Applied rewrites 36.0%

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

Alternative 13: 23.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 1\right)\right)}\right) \end{array} \]

FPCore

dY.v_m = (fabs.f32 dY.v)
(FPCore (w h d dX.u dX.v dX.w dY.u dY.v_m dY.w)
 :precision binary32
 (log2
  (sqrt
   (fmax (fma (floor w) dX.u (* (floor h) dX.v)) (fma (floor h) dY.v_m 1.0)))))

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	return log2f(sqrtf(fmaxf(fmaf(floorf(w), dX_46_u, (floorf(h) * dX_46_v)), fmaf(floorf(h), dY_46_v_m, 1.0f))));
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	return log2(sqrt(fmax(fma(floor(w), dX_46_u, Float32(floor(h) * dX_46_v)), fma(floor(h), dY_46_v_m, Float32(1.0)))))
end

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 49.9%

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

5. Applied rewrites 58.7%

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\color{blue}{\frac{\left|\left\lfloor w\right\rfloor \cdot dX.u\right| - \left\lfloor d\right\rfloor \cdot dX.w}{1} \cdot \frac{\mathsf{fma}\left(\left\lfloor d\right\rfloor , dX.w, \left|\left\lfloor w\right\rfloor \cdot dX.u\right|\right)}{\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor d\right\rfloor \cdot dX.w\right)}}, \left\lfloor w\right\rfloor \cdot dX.u - \left\lfloor d\right\rfloor \cdot dX.w, \left(\left\lfloor h\right\rfloor \cdot dX.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dX.v\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dY.u, \left\lfloor d\right\rfloor \cdot dY.w\right), \left\lfloor w\right\rfloor \cdot dY.u - \left\lfloor d\right\rfloor \cdot dY.w, \left(\left\lfloor h\right\rfloor \cdot dY.v\right) \cdot \left(\left\lfloor h\right\rfloor \cdot dY.v\right)\right)\right)}\right) \]

6. Taylor expanded in dX.w around 0

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

8. Applied rewrites 52.2%

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

10. Applied rewrites 21.9%

    \[\leadsto \color{blue}{\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left\lfloor w\right\rfloor , dX.u, \left\lfloor h\right\rfloor \cdot dX.v\right), \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v, 1\right)\right)}\right)} \]
11. Add Preprocessing

Alternative 14: 22.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\right) \end{array} \]

FPCore

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

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	return log2f(sqrtf(fmaxf(2.0f, fmaf((floorf(h) * floorf(h)), dY_46_v_m, 2.0f))));
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	return log2(sqrt(fmax(Float32(2.0), fma(Float32(floor(h) * floor(h)), dY_46_v_m, Float32(2.0)))))
end

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 49.9%

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

4. Taylor expanded in dX.v around 0

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

6. Applied rewrites 41.4%

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

8. Applied rewrites 22.9%

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

10. Applied rewrites 23.5%

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

Alternative 15: 21.9% accurate, 5.3× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v\_m, 2\right)\right)}\right) \end{array} \]

FPCore

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

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	return log2f(sqrtf(fmaxf(2.0f, fmaf(floorf(h), dY_46_v_m, 2.0f))));
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	return log2(sqrt(fmax(Float32(2.0), fma(floor(h), dY_46_v_m, Float32(2.0)))))
end

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 49.9%

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

4. Taylor expanded in dX.v around 0

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

6. Applied rewrites 41.4%

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

8. Applied rewrites 22.9%

   \[\leadsto \log_{2} \color{blue}{\left(\sqrt{\mathsf{max}\left(2, \mathsf{fma}\left(\left\lfloor h\right\rfloor , dY.v, 2\right)\right)}\right)} \]
9. Add Preprocessing

Alternative 16: 19.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} dY.v_m = \left|dY.v\right| \\ \log_{2} \left(\sqrt{\mathsf{max}\left(2, 2\right)}\right) \end{array} \]

FPCore

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

C

dY.v_m = fabs(dY_46_v);
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_m, float dY_46_w) {
	return log2f(sqrtf(fmaxf(2.0f, 2.0f)));
}

Julia

dY.v_m = abs(dY_46_v)
function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	return log2(sqrt(fmax(Float32(2.0), Float32(2.0))))
end

MATLAB

dY.v_m = abs(dY_46_v);
function tmp = code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v_m, dY_46_w)
	tmp = log2(sqrt(max(single(2.0), single(2.0))));
end

Derivation

1. Initial program 67.6%

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

3. Applied rewrites 49.9%

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

4. Taylor expanded in dX.v around 0

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

6. Applied rewrites 41.4%

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

8. Applied rewrites 22.9%

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

9. Taylor expanded in dY.v around 0

   \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(2, \color{blue}{2}\right)}\right) \]

11. Applied rewrites 19.5%

    \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(2, \color{blue}{2}\right)}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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)))))))