Isotropic LOD (LOD)

Percentage Accurate: 67.4% → 70.3%

Time: 39.2s

Alternatives: 6

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

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (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: 67.4% accurate, 1.0× speedup

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

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (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: 70.3% accurate, 0.5× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_1 := \left\lfloor h\right\rfloor \cdot dX.v\\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_4 := t\_3 \cdot t\_3\\ t_5 := t\_1 \cdot t\_1\\ t_6 := \left\lfloor h\right\rfloor \cdot dY.v\\ t_7 := t\_2 \cdot t\_2\\ t_8 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_9 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_10 := \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor \\ t_11 := \left\lfloor 0\right\rfloor \cdot dY.u\\ \mathbf{if}\;\mathsf{max}\left(\left(t\_9 \cdot t\_9 + t\_5\right) + t\_4, \left(t\_0 \cdot t\_0 + t\_6 \cdot t\_6\right) + t\_7\right) \leq 2.500000072189495 \cdot 10^{+38}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_9, t\_9, \mathsf{fma}\left(t\_10 \cdot dX.w, dX.w, t\_5\right)\right), \mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(t\_10 \cdot dY.w, dY.w, \left(dY.v \cdot dY.v\right) \cdot t\_8\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor , t\_4\right), \mathsf{fma}\left(t\_11, t\_11, \mathsf{fma}\left(dY.v \cdot dY.v, t\_8, t\_7\right)\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor w) dY.u))
       (t_1 (* (floor h) dX.v))
       (t_2 (* (floor d) dY.w))
       (t_3 (* (floor d) dX.w))
       (t_4 (* t_3 t_3))
       (t_5 (* t_1 t_1))
       (t_6 (* (floor h) dY.v))
       (t_7 (* t_2 t_2))
       (t_8 (* (floor h) (floor h)))
       (t_9 (* (floor w) dX.u))
       (t_10 (* (floor d) (floor d)))
       (t_11 (* (floor 0.0) dY.u)))
  (if (<=
       (fmax
        (+ (+ (* t_9 t_9) t_5) t_4)
        (+ (+ (* t_0 t_0) (* t_6 t_6)) t_7))
       2.500000072189495e+38)
    (log2
     (sqrt
      (fmax
       (fma t_9 t_9 (fma (* t_10 dX.w) dX.w t_5))
       (fma t_0 t_0 (fma (* t_10 dY.w) dY.w (* (* dY.v dY.v) t_8))))))
    (log2
     (sqrt
      (fmax
       (fma (* dX.u dX.u) (* (floor 0.0) (floor 0.0)) t_4)
       (fma t_11 t_11 (fma (* dY.v dY.v) t_8 t_7))))))))

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) * dX_46_v;
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = floorf(d) * dX_46_w;
	float t_4 = t_3 * t_3;
	float t_5 = t_1 * t_1;
	float t_6 = floorf(h) * dY_46_v;
	float t_7 = t_2 * t_2;
	float t_8 = floorf(h) * floorf(h);
	float t_9 = floorf(w) * dX_46_u;
	float t_10 = floorf(d) * floorf(d);
	float t_11 = floorf(0.0f) * dY_46_u;
	float tmp;
	if (fmaxf((((t_9 * t_9) + t_5) + t_4), (((t_0 * t_0) + (t_6 * t_6)) + t_7)) <= 2.500000072189495e+38f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_9, t_9, fmaf((t_10 * dX_46_w), dX_46_w, t_5)), fmaf(t_0, t_0, fmaf((t_10 * dY_46_w), dY_46_w, ((dY_46_v * dY_46_v) * t_8))))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), (floorf(0.0f) * floorf(0.0f)), t_4), fmaf(t_11, t_11, fmaf((dY_46_v * dY_46_v), t_8, t_7)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dY_46_u)
	t_1 = Float32(floor(h) * dX_46_v)
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(floor(d) * dX_46_w)
	t_4 = Float32(t_3 * t_3)
	t_5 = Float32(t_1 * t_1)
	t_6 = Float32(floor(h) * dY_46_v)
	t_7 = Float32(t_2 * t_2)
	t_8 = Float32(floor(h) * floor(h))
	t_9 = Float32(floor(w) * dX_46_u)
	t_10 = Float32(floor(d) * floor(d))
	t_11 = Float32(floor(Float32(0.0)) * dY_46_u)
	tmp = Float32(0.0)
	if (fmax(Float32(Float32(Float32(t_9 * t_9) + t_5) + t_4), Float32(Float32(Float32(t_0 * t_0) + Float32(t_6 * t_6)) + t_7)) <= Float32(2.500000072189495e+38))
		tmp = log2(sqrt(fmax(fma(t_9, t_9, fma(Float32(t_10 * dX_46_w), dX_46_w, t_5)), fma(t_0, t_0, fma(Float32(t_10 * dY_46_w), dY_46_w, Float32(Float32(dY_46_v * dY_46_v) * t_8))))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), Float32(floor(Float32(0.0)) * floor(Float32(0.0))), t_4), fma(t_11, t_11, fma(Float32(dY_46_v * dY_46_v), t_8, t_7)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fmax.f32 (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dX.u) (*.f32 (floor.f32 w) dX.u)) (*.f32 (*.f32 (floor.f32 h) dX.v) (*.f32 (floor.f32 h) dX.v))) (*.f32 (*.f32 (floor.f32 d) dX.w) (*.f32 (floor.f32 d) dX.w))) (+.f32 (+.f32 (*.f32 (*.f32 (floor.f32 w) dY.u) (*.f32 (floor.f32 w) dY.u)) (*.f32 (*.f32 (floor.f32 h) dY.v) (*.f32 (floor.f32 h) dY.v))) (*.f32 (*.f32 (floor.f32 d) dY.w) (*.f32 (floor.f32 d) dY.w)))) < 2.50000007e38

   1. Initial program 67.4%

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

   3. Applied rewrites 67.4%

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

   5. Applied rewrites 67.4%

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

   if 2.50000007e38 < (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 67.4%

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

   2. Taylor expanded in dX.v around 0

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

   4. Applied rewrites 60.3%

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

   6. Applied rewrites 60.3%

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

   7. Taylor expanded in undef-var around zero

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor , \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot dY.u, \left\lfloor 0\right\rfloor \cdot dY.u, \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)\right)}\right) \]

   9. Applied rewrites 44.4%

      \[\leadsto \log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, \left\lfloor 0\right\rfloor \cdot \left\lfloor 0\right\rfloor , \left(\left\lfloor d\right\rfloor \cdot dX.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dX.w\right)\right), \mathsf{fma}\left(\left\lfloor 0\right\rfloor \cdot dY.u, \left\lfloor 0\right\rfloor \cdot dY.u, \mathsf{fma}\left(dY.v \cdot dY.v, \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor , \left(\left\lfloor d\right\rfloor \cdot dY.w\right) \cdot \left(\left\lfloor d\right\rfloor \cdot dY.w\right)\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 65.3% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_2 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_3 := \mathsf{fma}\left(dY.v \cdot dY.v, t\_2, t\_1 \cdot t\_1\right)\\ t_4 := \left\lfloor w\right\rfloor \cdot \left|dY.u\right|\\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := \mathsf{fma}\left(dX.v \cdot dX.v, t\_2, t\_5 \cdot t\_5\right)\\ \mathbf{if}\;\left|dY.u\right| \leq 13985590:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, t\_0, t\_6\right), t\_3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(t\_6, \mathsf{fma}\left(t\_4, t\_4, t\_3\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor w) dX.u))
       (t_1 (* (floor d) dY.w))
       (t_2 (* (floor h) (floor h)))
       (t_3 (fma (* dY.v dY.v) t_2 (* t_1 t_1)))
       (t_4 (* (floor w) (fabs dY.u)))
       (t_5 (* (floor d) dX.w))
       (t_6 (fma (* dX.v dX.v) t_2 (* t_5 t_5))))
  (if (<= (fabs dY.u) 13985590.0)
    (log2 (sqrt (fmax (fma t_0 t_0 t_6) t_3)))
    (log2 (sqrt (fmax t_6 (fma t_4 t_4 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) * dX_46_u;
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = floorf(h) * floorf(h);
	float t_3 = fmaf((dY_46_v * dY_46_v), t_2, (t_1 * t_1));
	float t_4 = floorf(w) * fabsf(dY_46_u);
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = fmaf((dX_46_v * dX_46_v), t_2, (t_5 * t_5));
	float tmp;
	if (fabsf(dY_46_u) <= 13985590.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, t_0, t_6), t_3)));
	} else {
		tmp = log2f(sqrtf(fmaxf(t_6, fmaf(t_4, t_4, t_3))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(floor(h) * floor(h))
	t_3 = fma(Float32(dY_46_v * dY_46_v), t_2, Float32(t_1 * t_1))
	t_4 = Float32(floor(w) * abs(dY_46_u))
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = fma(Float32(dX_46_v * dX_46_v), t_2, Float32(t_5 * t_5))
	tmp = Float32(0.0)
	if (abs(dY_46_u) <= Float32(13985590.0))
		tmp = log2(sqrt(fmax(fma(t_0, t_0, t_6), t_3)));
	else
		tmp = log2(sqrt(fmax(t_6, fma(t_4, t_4, t_3))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.u < 13985590

   1. Initial program 67.4%

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

   2. Taylor expanded in dY.u around 0

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

   4. Applied rewrites 60.1%

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

   6. Applied rewrites 60.1%

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

   if 13985590 < dY.u

   1. Initial program 67.4%

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

   2. Taylor expanded in dX.u around 0

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

   4. Applied rewrites 60.1%

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

   6. Applied rewrites 60.1%

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

Alternative 3: 65.2% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor w\right\rfloor \cdot dX.u\\ t_1 := \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor \\ t_2 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_5 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_6 := t\_5 \cdot t\_5\\ t_7 := \left\lfloor w\right\rfloor \cdot dY.u\\ \mathbf{if}\;\left|dY.v\right| \leq 22541930:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(dX.v \cdot dX.v, t\_4, t\_6\right)\right), \mathsf{fma}\left(dY.u \cdot dY.u, t\_1, t\_3\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.u \cdot dX.u, t\_1, t\_6\right), \mathsf{fma}\left(t\_7, t\_7, \mathsf{fma}\left(\left|dY.v\right| \cdot \left|dY.v\right|, t\_4, t\_3\right)\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor w) dX.u))
       (t_1 (* (floor w) (floor w)))
       (t_2 (* (floor d) dY.w))
       (t_3 (* t_2 t_2))
       (t_4 (* (floor h) (floor h)))
       (t_5 (* (floor d) dX.w))
       (t_6 (* t_5 t_5))
       (t_7 (* (floor w) dY.u)))
  (if (<= (fabs dY.v) 22541930.0)
    (log2
     (sqrt
      (fmax
       (fma t_0 t_0 (fma (* dX.v dX.v) t_4 t_6))
       (fma (* dY.u dY.u) t_1 t_3))))
    (log2
     (sqrt
      (fmax
       (fma (* dX.u dX.u) t_1 t_6)
       (fma t_7 t_7 (fma (* (fabs dY.v) (fabs dY.v)) t_4 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) * dX_46_u;
	float t_1 = floorf(w) * floorf(w);
	float t_2 = floorf(d) * dY_46_w;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(h) * floorf(h);
	float t_5 = floorf(d) * dX_46_w;
	float t_6 = t_5 * t_5;
	float t_7 = floorf(w) * dY_46_u;
	float tmp;
	if (fabsf(dY_46_v) <= 22541930.0f) {
		tmp = log2f(sqrtf(fmaxf(fmaf(t_0, t_0, fmaf((dX_46_v * dX_46_v), t_4, t_6)), fmaf((dY_46_u * dY_46_u), t_1, t_3))));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_u * dX_46_u), t_1, t_6), fmaf(t_7, t_7, fmaf((fabsf(dY_46_v) * fabsf(dY_46_v)), t_4, t_3)))));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(w) * dX_46_u)
	t_1 = Float32(floor(w) * floor(w))
	t_2 = Float32(floor(d) * dY_46_w)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(floor(h) * floor(h))
	t_5 = Float32(floor(d) * dX_46_w)
	t_6 = Float32(t_5 * t_5)
	t_7 = Float32(floor(w) * dY_46_u)
	tmp = Float32(0.0)
	if (abs(dY_46_v) <= Float32(22541930.0))
		tmp = log2(sqrt(fmax(fma(t_0, t_0, fma(Float32(dX_46_v * dX_46_v), t_4, t_6)), fma(Float32(dY_46_u * dY_46_u), t_1, t_3))));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_u * dX_46_u), t_1, t_6), fma(t_7, t_7, fma(Float32(abs(dY_46_v) * abs(dY_46_v)), t_4, t_3)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dY.v < 22541930

   1. Initial program 67.4%

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

   2. Taylor expanded in dY.v around 0

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 60.6%

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

   if 22541930 < dY.v

   1. Initial program 67.4%

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

   2. Taylor expanded in dX.v around 0

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

   4. Applied rewrites 60.3%

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

   6. Applied rewrites 60.3%

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

Alternative 4: 65.1% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_2 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_3 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_4 := \mathsf{fma}\left(t\_3, t\_3, \mathsf{fma}\left(dY.v \cdot dY.v, t\_0, t\_1 \cdot t\_1\right)\right)\\ t_5 := \left\lfloor w\right\rfloor \cdot \left|dX.u\right|\\ \mathbf{if}\;\left|dX.u\right| \leq 1999.3817138671875:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, t\_0, t\_2 \cdot t\_2\right), t\_4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.w \cdot dX.w, \left\lfloor d\right\rfloor \cdot \left\lfloor d\right\rfloor , t\_5 \cdot t\_5\right), t\_4\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor h) (floor h)))
       (t_1 (* (floor d) dY.w))
       (t_2 (* (floor d) dX.w))
       (t_3 (* (floor w) dY.u))
       (t_4 (fma t_3 t_3 (fma (* dY.v dY.v) t_0 (* t_1 t_1))))
       (t_5 (* (floor w) (fabs dX.u))))
  (if (<= (fabs dX.u) 1999.3817138671875)
    (log2 (sqrt (fmax (fma (* dX.v dX.v) t_0 (* t_2 t_2)) t_4)))
    (log2
     (sqrt
      (fmax
       (fma (* dX.w dX.w) (* (floor d) (floor d)) (* t_5 t_5))
       t_4))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * floorf(h);
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = floorf(d) * dX_46_w;
	float t_3 = floorf(w) * dY_46_u;
	float t_4 = fmaf(t_3, t_3, fmaf((dY_46_v * dY_46_v), t_0, (t_1 * t_1)));
	float t_5 = floorf(w) * fabsf(dX_46_u);
	float tmp;
	if (fabsf(dX_46_u) <= 1999.3817138671875f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_v * dX_46_v), t_0, (t_2 * t_2)), t_4)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_w * dX_46_w), (floorf(d) * floorf(d)), (t_5 * t_5)), t_4)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * floor(h))
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(floor(d) * dX_46_w)
	t_3 = Float32(floor(w) * dY_46_u)
	t_4 = fma(t_3, t_3, fma(Float32(dY_46_v * dY_46_v), t_0, Float32(t_1 * t_1)))
	t_5 = Float32(floor(w) * abs(dX_46_u))
	tmp = Float32(0.0)
	if (abs(dX_46_u) <= Float32(1999.3817138671875))
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_v * dX_46_v), t_0, Float32(t_2 * t_2)), t_4)));
	else
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_w * dX_46_w), Float32(floor(d) * floor(d)), Float32(t_5 * t_5)), t_4)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1999.38171

   1. Initial program 67.4%

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

   2. Taylor expanded in dX.u around 0

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

   4. Applied rewrites 60.1%

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

   6. Applied rewrites 60.1%

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

   if 1999.38171 < dX.u

   1. Initial program 67.4%

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

   2. Taylor expanded in dX.v around 0

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

   4. Applied rewrites 60.3%

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

   6. Applied rewrites 60.3%

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

   8. Applied rewrites 60.3%

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

Alternative 5: 65.1% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} t_0 := \left\lfloor h\right\rfloor \cdot \left\lfloor h\right\rfloor \\ t_1 := \left\lfloor d\right\rfloor \cdot dY.w\\ t_2 := \left\lfloor d\right\rfloor \cdot dX.w\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \left\lfloor w\right\rfloor \cdot dY.u\\ t_5 := \mathsf{fma}\left(t\_4, t\_4, \mathsf{fma}\left(dY.v \cdot dY.v, t\_0, t\_1 \cdot t\_1\right)\right)\\ \mathbf{if}\;\left|dX.u\right| \leq 1999.3817138671875:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(dX.v \cdot dX.v, t\_0, t\_3\right), t\_5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log_{2} \left(\sqrt{\mathsf{max}\left(\mathsf{fma}\left(\left|dX.u\right| \cdot \left|dX.u\right|, \left\lfloor w\right\rfloor \cdot \left\lfloor w\right\rfloor , t\_3\right), t\_5\right)}\right)\\ \end{array} \]

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor h) (floor h)))
       (t_1 (* (floor d) dY.w))
       (t_2 (* (floor d) dX.w))
       (t_3 (* t_2 t_2))
       (t_4 (* (floor w) dY.u))
       (t_5 (fma t_4 t_4 (fma (* dY.v dY.v) t_0 (* t_1 t_1)))))
  (if (<= (fabs dX.u) 1999.3817138671875)
    (log2 (sqrt (fmax (fma (* dX.v dX.v) t_0 t_3) t_5)))
    (log2
     (sqrt
      (fmax
       (fma (* (fabs dX.u) (fabs dX.u)) (* (floor w) (floor w)) t_3)
       t_5))))))

C

float code(float w, float h, float d, float dX_46_u, float dX_46_v, float dX_46_w, float dY_46_u, float dY_46_v, float dY_46_w) {
	float t_0 = floorf(h) * floorf(h);
	float t_1 = floorf(d) * dY_46_w;
	float t_2 = floorf(d) * dX_46_w;
	float t_3 = t_2 * t_2;
	float t_4 = floorf(w) * dY_46_u;
	float t_5 = fmaf(t_4, t_4, fmaf((dY_46_v * dY_46_v), t_0, (t_1 * t_1)));
	float tmp;
	if (fabsf(dX_46_u) <= 1999.3817138671875f) {
		tmp = log2f(sqrtf(fmaxf(fmaf((dX_46_v * dX_46_v), t_0, t_3), t_5)));
	} else {
		tmp = log2f(sqrtf(fmaxf(fmaf((fabsf(dX_46_u) * fabsf(dX_46_u)), (floorf(w) * floorf(w)), t_3), t_5)));
	}
	return tmp;
}

Julia

function code(w, h, d, dX_46_u, dX_46_v, dX_46_w, dY_46_u, dY_46_v, dY_46_w)
	t_0 = Float32(floor(h) * floor(h))
	t_1 = Float32(floor(d) * dY_46_w)
	t_2 = Float32(floor(d) * dX_46_w)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(floor(w) * dY_46_u)
	t_5 = fma(t_4, t_4, fma(Float32(dY_46_v * dY_46_v), t_0, Float32(t_1 * t_1)))
	tmp = Float32(0.0)
	if (abs(dX_46_u) <= Float32(1999.3817138671875))
		tmp = log2(sqrt(fmax(fma(Float32(dX_46_v * dX_46_v), t_0, t_3), t_5)));
	else
		tmp = log2(sqrt(fmax(fma(Float32(abs(dX_46_u) * abs(dX_46_u)), Float32(floor(w) * floor(w)), t_3), t_5)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if dX.u < 1999.38171

   1. Initial program 67.4%

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

   2. Taylor expanded in dX.u around 0

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

   4. Applied rewrites 60.1%

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

   6. Applied rewrites 60.1%

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

   if 1999.38171 < dX.u

   1. Initial program 67.4%

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

   2. Taylor expanded in dX.v around 0

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

   4. Applied rewrites 60.3%

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

   6. Applied rewrites 60.3%

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

Alternative 6: 60.3% accurate, 1.2× speedup

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

Math

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

FPCore

(FPCore (w h d dX.u dX.v dX.w dY.u dY.v dY.w)
  :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)))
  (let* ((t_0 (* (floor d) dY.w))
       (t_1 (* (floor d) dX.w))
       (t_2 (* (floor w) dY.u)))
  (log2
   (sqrt
    (fmax
     (fma (* dX.u dX.u) (* (floor w) (floor w)) (* t_1 t_1))
     (fma
      t_2
      t_2
      (fma (* dY.v dY.v) (* (floor h) (floor h)) (* t_0 t_0))))))))

C

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

Julia

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

Derivation

1. Initial program 67.4%

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

2. Taylor expanded in dX.v around 0

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

4. Applied rewrites 60.3%

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

6. Applied rewrites 60.3%

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

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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)))))))