UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 7.2s

Alternatives: 26

Speedup: 1.2×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0))))
       (t_2 (* (* uy 2.0) PI)))
  (+
   (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
       (t_1 (sqrt (- 1.0 (* t_0 t_0))))
       (t_2 (* (* uy 2.0) PI)))
  (+
   (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
   (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 99.0% accurate, 1.0× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\\ t_1 := \pi \cdot \left(uy + uy\right)\\ t_2 := \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, t\_0 \cdot ux, 1\right)}\\ \mathsf{fma}\left(yi \cdot t\_2, \sin t\_1, \mathsf{fma}\left(xi \cdot t\_2, \cos t\_1, zi \cdot t\_0\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* (* maxCos (- 1.0 ux)) ux))
       (t_1 (* PI (+ uy uy)))
       (t_2 (sqrt (fma (* (- ux 1.0) maxCos) (* t_0 ux) 1.0))))
  (fma (* yi t_2) (sin t_1) (fma (* xi t_2) (cos t_1) (* zi t_0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (maxCos * (1.0f - ux)) * ux;
	float t_1 = ((float) M_PI) * (uy + uy);
	float t_2 = sqrtf(fmaf(((ux - 1.0f) * maxCos), (t_0 * ux), 1.0f));
	return fmaf((yi * t_2), sinf(t_1), fmaf((xi * t_2), cosf(t_1), (zi * t_0)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux)
	t_1 = Float32(Float32(pi) * Float32(uy + uy))
	t_2 = sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(t_0 * ux), Float32(1.0)))
	return fma(Float32(yi * t_2), sin(t_1), fma(Float32(xi * t_2), cos(t_1), Float32(zi * t_0)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(yi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \sin \left(\pi \cdot \left(uy + uy\right)\right), \mathsf{fma}\left(xi \cdot \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)}, \cos \left(\pi \cdot \left(uy + uy\right)\right), zi \cdot \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)\right)\right)} \]
4. Add Preprocessing

Alternative 2: 99.0% accurate, 1.2× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\left(-\mathsf{fma}\left(uy + uy, \pi, 1.5707963705062866\right)\right) + 1.5707963705062866\right) \cdot xi\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma
 (- 1.0 ux)
 (* (* zi ux) maxCos)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (fma
   (sin (* PI (+ uy uy)))
   yi
   (*
    (cos
     (+ (- (fma (+ uy uy) PI 1.5707963705062866)) 1.5707963705062866))
    xi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(sinf((((float) M_PI) * (uy + uy))), yi, (cosf((-fmaf((uy + uy), ((float) M_PI), 1.5707963705062866f) + 1.5707963705062866f)) * xi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(sin(Float32(Float32(pi) * Float32(uy + uy))), yi, Float32(cos(Float32(Float32(-fma(Float32(uy + uy), Float32(pi), Float32(1.5707963705062866))) + Float32(1.5707963705062866))) * xi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   7. lower-*.f32 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]
6. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \color{blue}{\cos \left(\pi \cdot \left(uy + uy\right)\right)} \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right)} \cdot xi\right)\right) \]

   3. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \color{blue}{\left(uy + uy\right)}\right) \cdot xi\right)\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \color{blue}{\left(uy \cdot \pi + uy \cdot \pi\right)} \cdot xi\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\color{blue}{uy \cdot \pi} + uy \cdot \pi\right) \cdot xi\right)\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(uy \cdot \pi + \color{blue}{uy \cdot \pi}\right) \cdot xi\right)\right) \]

   7. cos-sum-rev N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \color{blue}{\left(\cos \left(uy \cdot \pi\right) \cdot \cos \left(uy \cdot \pi\right) - \sin \left(uy \cdot \pi\right) \cdot \sin \left(uy \cdot \pi\right)\right)} \cdot xi\right)\right) \]

   8. sub-negate-rev N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \color{blue}{\left(\mathsf{neg}\left(\left(\sin \left(uy \cdot \pi\right) \cdot \sin \left(uy \cdot \pi\right) - \cos \left(uy \cdot \pi\right) \cdot \cos \left(uy \cdot \pi\right)\right)\right)\right)} \cdot xi\right)\right) \]

   9. sub-negate N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\cos \left(uy \cdot \pi\right) \cdot \cos \left(uy \cdot \pi\right) - \sin \left(uy \cdot \pi\right) \cdot \sin \left(uy \cdot \pi\right)\right)\right)\right)}\right)\right) \cdot xi\right)\right) \]

   10. cos-sum-rev N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\cos \left(uy \cdot \pi + uy \cdot \pi\right)}\right)\right)\right)\right) \cdot xi\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\color{blue}{uy \cdot \pi} + uy \cdot \pi\right)\right)\right)\right)\right) \cdot xi\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(uy \cdot \pi + \color{blue}{uy \cdot \pi}\right)\right)\right)\right)\right) \cdot xi\right)\right) \]

   13. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right)}\right)\right)\right)\right) \cdot xi\right)\right) \]

   14. lift-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \left(\pi \cdot \color{blue}{\left(uy + uy\right)}\right)\right)\right)\right)\right) \cdot xi\right)\right) \]

   15. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\cos \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right)}\right)\right)\right)\right) \cdot xi\right)\right) \]

   16. sin-+PI/2-rev N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\sin \left(\pi \cdot \left(uy + uy\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right)\right)\right)\right) \cdot xi\right)\right) \]

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \color{blue}{\cos \left(\left(-\mathsf{fma}\left(uy + uy, \pi, \pi \cdot 0.5\right)\right) + \pi \cdot 0.5\right)} \cdot xi\right)\right) \]

8. Evaluated real constant 98.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\left(-\mathsf{fma}\left(uy + uy, \pi, \color{blue}{1.5707963705062866}\right)\right) + \pi \cdot 0.5\right) \cdot xi\right)\right) \]

9. Evaluated real constant 98.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\left(-\mathsf{fma}\left(uy + uy, \pi, 1.5707963705062866\right)\right) + \color{blue}{1.5707963705062866}\right) \cdot xi\right)\right) \]
10. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(1 - ux, maxCos \cdot \left(zi \cdot ux\right), \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin t\_0, yi, \cos t\_0 \cdot xi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* PI (+ uy uy))))
  (fma
   (- 1.0 ux)
   (* maxCos (* zi ux))
   (*
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0))
    (fma (sin t_0) yi (* (cos t_0) xi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((1.0f - ux), (maxCos * (zi * ux)), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(sinf(t_0), yi, (cosf(t_0) * xi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(Float32(1.0) - ux), Float32(maxCos * Float32(zi * ux)), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(sin(t_0), yi, Float32(cos(t_0) * xi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, maxCos \cdot \left(zi \cdot ux\right), \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Add Preprocessing

Alternative 4: 98.8% accurate, 1.3× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin t\_0, yi, \cos t\_0 \cdot xi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* PI (+ uy uy))))
  (fma
   (- 1.0 ux)
   (* (* zi ux) maxCos)
   (*
    (sqrt (fma (* (- ux 1.0) maxCos) (* (* maxCos ux) ux) 1.0))
    (fma (sin t_0) yi (* (cos t_0) xi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), ((maxCos * ux) * ux), 1.0f)) * fmaf(sinf(t_0), yi, (cosf(t_0) * xi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(maxCos * ux) * ux), Float32(1.0))) * fma(sin(t_0), yi, Float32(cos(t_0) * xi))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   7. lower-*.f32 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

6. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.8%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]
9. Add Preprocessing

Alternative 5: 98.7% accurate, 1.6× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(\pi + \pi\right) \cdot uy\\ \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \mathsf{fma}\left(\sin t\_0, yi, \cos t\_0 \cdot xi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* (+ PI PI) uy)))
  (fma
   (- 1.0 ux)
   (* (* zi ux) maxCos)
   (fma (sin t_0) yi (* (cos t_0) xi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) + ((float) M_PI)) * uy;
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), fmaf(sinf(t_0), yi, (cosf(t_0) * xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) + Float32(pi)) * uy)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), fma(sin(t_0), yi, Float32(cos(t_0) * xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot maxCos + \mathsf{fma}\left(\color{blue}{xi}, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot maxCos + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \left(\left(1 - ux\right) \cdot \left(zi \cdot ux\right)\right) \cdot maxCos + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \left(\left(1 - ux\right) \cdot \left(zi \cdot ux\right)\right) \cdot maxCos + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \left(1 - ux\right) \cdot \left(\left(zi \cdot ux\right) \cdot maxCos\right) + \mathsf{fma}\left(\color{blue}{xi}, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \left(1 - ux\right) \cdot \left(\left(zi \cdot ux\right) \cdot maxCos\right) + \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   11. lower-fma.f32 98.7%

       \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   12. lift-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   13. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \mathsf{fma}\left(\sin \left(\left(\pi + \pi\right) \cdot uy\right), yi, \cos \left(\left(\pi + \pi\right) \cdot uy\right) \cdot xi\right)\right) \]
10. Add Preprocessing

Alternative 6: 98.2% accurate, 1.6× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.008500000461935997:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* 2.0 (* uy PI))))
  (if (<= uy 0.008500000461935997)
    (+
     xi
     (fma
      maxCos
      (* ux (* zi (- 1.0 ux)))
      (*
       uy
       (fma
        2.0
        (* yi PI)
        (*
         uy
         (fma
          -2.0
          (* xi (pow PI 2.0))
          (* -1.3333333333333333 (* uy (* yi (pow PI 3.0))))))))))
    (fma maxCos (* ux zi) (fma xi (cos t_0) (* yi (sin t_0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (uy <= 0.008500000461935997f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * powf(((float) M_PI), 2.0f)), (-1.3333333333333333f * (uy * (yi * powf(((float) M_PI), 3.0f)))))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), fmaf(xi, cosf(t_0), (yi * sinf(t_0))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.008500000461935997))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))))))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), fma(xi, cos(t_0), Float32(yi * sin(t_0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00850000046

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\pi}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lower--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto xi + \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]

   if 0.00850000046 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{zi}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot zi, \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.9% accurate, 1.7× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.10849999636411667:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi}{yi}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (if (<= uy 0.10849999636411667)
  (+
   xi
   (fma
    maxCos
    (* ux (* zi (- 1.0 ux)))
    (*
     uy
     (fma
      2.0
      (* yi PI)
      (*
       uy
       (fma
        -2.0
        (* xi (pow PI 2.0))
        (* -1.3333333333333333 (* uy (* yi (pow PI 3.0))))))))))
  (fma
   (- 1.0 ux)
   (* (* zi ux) maxCos)
   (*
    (sqrt
     (fma
      (* (- ux 1.0) maxCos)
      (* (* (* maxCos (- 1.0 ux)) ux) ux)
      1.0))
    (* yi (+ (sin (* 2.0 (* uy PI))) (/ xi yi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.10849999636411667f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * powf(((float) M_PI), 2.0f)), (-1.3333333333333333f * (uy * (yi * powf(((float) M_PI), 3.0f)))))))));
	} else {
		tmp = fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * (yi * (sinf((2.0f * (uy * ((float) M_PI)))) + (xi / yi)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.10849999636411667))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))))))))));
	else
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(yi * Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) + Float32(xi / yi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.108499996

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\pi}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lower--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto xi + \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]

   if 0.108499996 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      7. lower-*.f32 99.0%

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. Taylor expanded in yi around inf

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{yi}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{yi}\right)}\right)\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{yi}}\right)\right)\right) \]

      3. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}}{yi}\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{\color{blue}{xi} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{yi}\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{yi}\right)\right)\right) \]

      6. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{yi}\right)\right)\right) \]

      7. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{yi}}\right)\right)\right) \]

   8. Applied rewrites 98.3%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{yi}\right)\right)}\right) \]

   9. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi}{\color{blue}{yi}}\right)\right)\right) \]
   10. Step-by-step derivation

       1. lower-/.f32 88.3%

          \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi}{yi}\right)\right)\right) \]

   11. Applied rewrites 88.3%

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + \frac{xi}{\color{blue}{yi}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.6% accurate, 1.8× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.10849999636411667:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot zi, ux, \left(\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot yi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (if (<= uy 0.10849999636411667)
  (+
   xi
   (fma
    maxCos
    (* ux (* zi (- 1.0 ux)))
    (*
     uy
     (fma
      2.0
      (* yi PI)
      (*
       uy
       (fma
        -2.0
        (* xi (pow PI 2.0))
        (* -1.3333333333333333 (* uy (* yi (pow PI 3.0))))))))))
  (fma
   (* (* maxCos (- 1.0 ux)) zi)
   ux
   (*
    (* (sin (* (+ uy uy) PI)) yi)
    (sqrt
     (fma
      (* (* (* (- ux 1.0) maxCos) ux) maxCos)
      (* (- 1.0 ux) ux)
      1.0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.10849999636411667f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * powf(((float) M_PI), 2.0f)), (-1.3333333333333333f * (uy * (yi * powf(((float) M_PI), 3.0f)))))))));
	} else {
		tmp = fmaf(((maxCos * (1.0f - ux)) * zi), ux, ((sinf(((uy + uy) * ((float) M_PI))) * yi) * sqrtf(fmaf(((((ux - 1.0f) * maxCos) * ux) * maxCos), ((1.0f - ux) * ux), 1.0f))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.10849999636411667))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))))))))));
	else
		tmp = fma(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * zi), ux, Float32(Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * yi) * sqrt(fma(Float32(Float32(Float32(Float32(ux - Float32(1.0)) * maxCos) * ux) * maxCos), Float32(Float32(Float32(1.0) - ux) * ux), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.108499996

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\pi}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lower--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto xi + \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]

   if 0.108499996 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]

   4. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-PI.f32 44.8%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 44.8%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

   8. Applied rewrites 44.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot zi, ux, \left(\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot yi\right) \cdot \sqrt{\mathsf{fma}\left(\left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot maxCos, \left(1 - ux\right) \cdot ux, 1\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.6% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.10849999636411667:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (if (<= uy 0.10849999636411667)
  (+
   xi
   (fma
    maxCos
    (* ux (* zi (- 1.0 ux)))
    (*
     uy
     (fma
      2.0
      (* yi PI)
      (*
       uy
       (fma
        -2.0
        (* xi (pow PI 2.0))
        (* -1.3333333333333333 (* uy (* yi (pow PI 3.0))))))))))
  (fma
   (- 1.0 ux)
   (* (* maxCos ux) zi)
   (*
    (sqrt (fma (* (- ux 1.0) maxCos) (* (* maxCos ux) ux) 1.0))
    (* yi (sin (* 2.0 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.10849999636411667f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * powf(((float) M_PI), 2.0f)), (-1.3333333333333333f * (uy * (yi * powf(((float) M_PI), 3.0f)))))))));
	} else {
		tmp = fmaf((1.0f - ux), ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), ((maxCos * ux) * ux), 1.0f)) * (yi * sinf((2.0f * (uy * ((float) M_PI)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.10849999636411667))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * (Float32(pi) ^ Float32(2.0))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * (Float32(pi) ^ Float32(3.0)))))))))));
	else
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(maxCos * ux) * ux), Float32(1.0))) * Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.108499996

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\pi}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto xi + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lower--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   10. Applied rewrites 89.6%

       \[\leadsto xi + \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot {\pi}^{2}, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot {\pi}^{3}\right)\right)\right)\right)\right)} \]

   if 0.108499996 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]

   4. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-PI.f32 44.8%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 44.8%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 44.7%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.6% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(ux - 1\right) \cdot maxCos\\ \mathbf{if}\;uy \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(t\_0, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(t\_0, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* (- ux 1.0) maxCos)))
  (if (<= uy 0.009999999776482582)
    (fma
     (- 1.0 ux)
     (* (* zi ux) maxCos)
     (*
      (sqrt (fma t_0 (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0))
      (+
       xi
       (*
        uy
        (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI)))))))
    (fma
     (- 1.0 ux)
     (* (* maxCos ux) zi)
     (*
      (sqrt (fma t_0 (* (* maxCos ux) ux) 1.0))
      (* yi (sin (* 2.0 (* uy PI)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (ux - 1.0f) * maxCos;
	float tmp;
	if (uy <= 0.009999999776482582f) {
		tmp = fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(t_0, (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * (xi + (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI))))))));
	} else {
		tmp = fmaf((1.0f - ux), ((maxCos * ux) * zi), (sqrtf(fmaf(t_0, ((maxCos * ux) * ux), 1.0f)) * (yi * sinf((2.0f * (uy * ((float) M_PI)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(ux - Float32(1.0)) * maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.009999999776482582))
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(t_0, Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi))))))));
	else
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(t_0, Float32(Float32(maxCos * ux) * ux), Float32(1.0))) * Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00999999978

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      2. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

      7. lower-*.f32 99.0%

         \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, \color{blue}{uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \color{blue}{\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. lower-PI.f32 86.1%

          \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   8. Applied rewrites 86.1%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

   if 0.00999999978 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]

   4. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-PI.f32 44.8%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 44.8%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 44.7%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.4% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \left(ux - 1\right) \cdot maxCos\\ t_1 := \left(maxCos \cdot ux\right) \cdot zi\\ \mathbf{if}\;uy \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, t\_1, \sqrt{\mathsf{fma}\left(t\_0, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, t\_1, \sqrt{\mathsf{fma}\left(t\_0, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* (- ux 1.0) maxCos)) (t_1 (* (* maxCos ux) zi)))
  (if (<= uy 0.009999999776482582)
    (fma
     (- 1.0 ux)
     t_1
     (*
      (sqrt (fma t_0 (* (* (* maxCos (- 1.0 ux)) ux) ux) 1.0))
      (+
       xi
       (*
        uy
        (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI)))))))
    (fma
     (- 1.0 ux)
     t_1
     (*
      (sqrt (fma t_0 (* (* maxCos ux) ux) 1.0))
      (* yi (sin (* 2.0 (* uy PI)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (ux - 1.0f) * maxCos;
	float t_1 = (maxCos * ux) * zi;
	float tmp;
	if (uy <= 0.009999999776482582f) {
		tmp = fmaf((1.0f - ux), t_1, (sqrtf(fmaf(t_0, (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * (xi + (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI))))))));
	} else {
		tmp = fmaf((1.0f - ux), t_1, (sqrtf(fmaf(t_0, ((maxCos * ux) * ux), 1.0f)) * (yi * sinf((2.0f * (uy * ((float) M_PI)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(ux - Float32(1.0)) * maxCos)
	t_1 = Float32(Float32(maxCos * ux) * zi)
	tmp = Float32(0.0)
	if (uy <= Float32(0.009999999776482582))
		tmp = fma(Float32(Float32(1.0) - ux), t_1, Float32(sqrt(fma(t_0, Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi))))))));
	else
		tmp = fma(Float32(Float32(1.0) - ux), t_1, Float32(sqrt(fma(t_0, Float32(Float32(maxCos * ux) * ux), Float32(1.0))) * Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00999999978

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]

   4. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, \color{blue}{uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \color{blue}{\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. lower-PI.f32 86.1%

          \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 86.1%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

   if 0.00999999978 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]

   4. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-PI.f32 44.8%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 44.8%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 44.7%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 87.4% accurate, 1.9× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (if (<= uy 0.009999999776482582)
  (fma
   maxCos
   (* ux (* zi (- 1.0 ux)))
   (+
    xi
    (* uy (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI))))))
  (fma
   (- 1.0 ux)
   (* (* maxCos ux) zi)
   (*
    (sqrt (fma (* (- ux 1.0) maxCos) (* (* maxCos ux) ux) 1.0))
    (* yi (sin (* 2.0 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.009999999776482582f) {
		tmp = fmaf(maxCos, (ux * (zi * (1.0f - ux))), (xi + (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf((1.0f - ux), ((maxCos * ux) * zi), (sqrtf(fmaf(((ux - 1.0f) * maxCos), ((maxCos * ux) * ux), 1.0f)) * (yi * sinf((2.0f * (uy * ((float) M_PI)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.009999999776482582))
		tmp = fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(Float32(Float32(1.0) - ux), Float32(Float32(maxCos * ux) * zi), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(maxCos * ux) * ux), Float32(1.0))) * Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00999999978

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f32 85.9%

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 85.9%

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   if 0.00999999978 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]

   4. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. lower-PI.f32 44.8%

         \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]

   6. Applied rewrites 44.8%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 44.7%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 87.2% accurate, 2.6× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\\ \mathbf{if}\;uy \leq 0.009999999776482582:\\ \;\;\;\;\mathsf{fma}\left(maxCos, t\_0, xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, t\_0, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* ux (* zi (- 1.0 ux)))))
  (if (<= uy 0.009999999776482582)
    (fma
     maxCos
     t_0
     (+
      xi
      (* uy (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI))))))
    (fma maxCos t_0 (* yi (sin (* 2.0 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * (zi * (1.0f - ux));
	float tmp;
	if (uy <= 0.009999999776482582f) {
		tmp = fmaf(maxCos, t_0, (xi + (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf(maxCos, t_0, (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.009999999776482582))
		tmp = fma(maxCos, t_0, Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, t_0, Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00999999978

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f32 85.9%

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 85.9%

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   if 0.00999999978 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in xi around 0

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-PI.f32 44.7%

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 44.7%

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 87.2% accurate, 2.6× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\\ \mathbf{if}\;uy \leq 0.30000001192092896:\\ \;\;\;\;\mathsf{fma}\left(maxCos, t\_0, xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, t\_0, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (let* ((t_0 (* ux (* zi (- 1.0 ux)))))
  (if (<= uy 0.30000001192092896)
    (fma
     maxCos
     t_0
     (+
      xi
      (* uy (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI))))))
    (fma maxCos t_0 (* xi (cos (* 2.0 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * (zi * (1.0f - ux));
	float tmp;
	if (uy <= 0.30000001192092896f) {
		tmp = fmaf(maxCos, t_0, (xi + (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI)))))));
	} else {
		tmp = fmaf(maxCos, t_0, (xi * cosf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.30000001192092896))
		tmp = fma(maxCos, t_0, Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, t_0, Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.300000012

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-+.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-pow.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lower-PI.f32 85.9%

          \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 85.9%

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

   if 0.300000012 < uy

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      6. lower-sqrt.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

      7. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

   8. Taylor expanded in yi around 0

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. lower--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-cos.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-PI.f32 59.1%

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 59.1%

       \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 85.9% accurate, 2.9× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma
 maxCos
 (* ux (* zi (- 1.0 ux)))
 (+
  xi
  (* uy (fma -2.0 (* uy (* xi (pow PI 2.0))) (* 2.0 (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(maxCos, (ux * (zi * (1.0f - ux))), (xi + (uy * fmaf(-2.0f, (uy * (xi * powf(((float) M_PI), 2.0f))), (2.0f * (yi * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(xi + Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-pow.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lower-PI.f32 85.9%

       \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]

10. Applied rewrites 85.9%

    \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi + uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot {\pi}^{2}\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right) \]
11. Add Preprocessing

Alternative 16: 82.0% accurate, 3.0× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(uy + uy, yi \cdot \pi, xi\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma
 (- 1.0 ux)
 (* (* zi ux) maxCos)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (fma (+ uy uy) (* yi PI) xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf((uy + uy), (yi * ((float) M_PI)), xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(Float32(uy + uy), Float32(yi * Float32(pi)), xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   7. lower-*.f32 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

6. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   5. lower-PI.f32 81.9%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

8. Applied rewrites 81.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \color{blue}{xi}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + xi\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + xi\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   6. count-2 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   7. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   8. lower-fma.f32 82.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(uy + uy, \color{blue}{yi \cdot \pi}, xi\right)\right) \]

10. Applied rewrites 82.0%

    \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(uy + uy, \color{blue}{yi \cdot \pi}, xi\right)\right) \]
11. Add Preprocessing

Alternative 17: 82.0% accurate, 3.0× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot uy, yi, xi\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma
 (- 1.0 ux)
 (* (* zi ux) maxCos)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (fma (* (+ PI PI) uy) yi xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(((((float) M_PI) + ((float) M_PI)) * uy), yi, xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(Float32(Float32(Float32(pi) + Float32(pi)) * uy), yi, xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   7. lower-*.f32 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

6. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   5. lower-PI.f32 81.9%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

8. Applied rewrites 81.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \color{blue}{xi}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + xi\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + xi\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   6. count-2 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   7. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(\pi \cdot yi\right) + xi\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(\left(uy + uy\right) \cdot \pi\right) \cdot yi + xi\right)\right) \]

   11. lift-+.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(\left(uy + uy\right) \cdot \pi\right) \cdot yi + xi\right)\right) \]

   12. count-2 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot yi + xi\right)\right) \]

   13. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + xi\right)\right) \]

   14. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + xi\right)\right) \]

   15. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + xi\right)\right) \]

   16. lower-fma.f32 82.0%

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(2 \cdot \left(uy \cdot \pi\right), \color{blue}{yi}, xi\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(2 \cdot \left(uy \cdot \pi\right), yi, xi\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(2 \cdot \left(uy \cdot \pi\right), yi, xi\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(2 \cdot \left(\pi \cdot uy\right), yi, xi\right)\right) \]

   20. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot \pi\right) \cdot uy, yi, xi\right)\right) \]

   21. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(2 \cdot \pi\right) \cdot uy, yi, xi\right)\right) \]

   22. count-2-rev N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot uy, yi, xi\right)\right) \]

   23. lower-+.f32 82.0%

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot uy, yi, xi\right)\right) \]

10. Applied rewrites 82.0%

    \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot uy, \color{blue}{yi}, xi\right)\right) \]
11. Add Preprocessing

Alternative 18: 81.9% accurate, 3.0× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(uy + uy\right) \cdot yi, \pi, xi\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma
 (- 1.0 ux)
 (* (* zi ux) maxCos)
 (*
  (sqrt
   (fma
    (* (- ux 1.0) maxCos)
    (* (* (* maxCos (- 1.0 ux)) ux) ux)
    1.0))
  (fma (* (+ uy uy) yi) PI xi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), (((maxCos * (1.0f - ux)) * ux) * ux), 1.0f)) * fmaf(((uy + uy) * yi), ((float) M_PI), xi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * ux) * ux), Float32(1.0))) * fma(Float32(Float32(uy + uy) * yi), Float32(pi), xi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   7. lower-*.f32 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

6. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   5. lower-PI.f32 81.9%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

8. Applied rewrites 81.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + \color{blue}{xi}\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + xi\right)\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + xi\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(2 \cdot uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   6. count-2 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   7. lift-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \left(yi \cdot \pi\right) + xi\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(\left(\left(uy + uy\right) \cdot yi\right) \cdot \pi + xi\right)\right) \]

   10. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(uy + uy\right) \cdot yi, \color{blue}{\pi}, xi\right)\right) \]

   11. lower-*.f32 81.9%

       \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(uy + uy\right) \cdot yi, \pi, xi\right)\right) \]

10. Applied rewrites 81.9%

    \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\left(uy + uy\right) \cdot yi, \color{blue}{\pi}, xi\right)\right) \]
11. Add Preprocessing

Alternative 19: 81.8% accurate, 3.3× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(maxCos \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma
 (- 1.0 ux)
 (* (* zi ux) maxCos)
 (*
  (sqrt (fma (* (- ux 1.0) maxCos) (* (* maxCos ux) ux) 1.0))
  (+ xi (* 2.0 (* uy (* yi PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((1.0f - ux), ((zi * ux) * maxCos), (sqrtf(fmaf(((ux - 1.0f) * maxCos), ((maxCos * ux) * ux), 1.0f)) * (xi + (2.0f * (uy * (yi * ((float) M_PI)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(1.0) - ux), Float32(Float32(zi * ux) * maxCos), Float32(sqrt(fma(Float32(Float32(ux - Float32(1.0)) * maxCos), Float32(Float32(maxCos * ux) * ux), Float32(1.0))) * Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi)))))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 - ux, \left(maxCos \cdot ux\right) \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right) \cdot zi}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(maxCos \cdot ux\right)} \cdot zi, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{maxCos \cdot \left(ux \cdot zi\right)}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(ux \cdot zi\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

   7. lower-*.f32 99.0%

      \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right)} \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

5. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(1 - ux, \color{blue}{\left(zi \cdot ux\right) \cdot maxCos}, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \left(uy + uy\right)\right), yi, \cos \left(\pi \cdot \left(uy + uy\right)\right) \cdot xi\right)\right) \]

6. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]
7. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   5. lower-PI.f32 81.9%

      \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]

8. Applied rewrites 81.9%

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot ux, 1\right)} \cdot \color{blue}{\left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)}\right) \]

9. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 81.8%

    \[\leadsto \mathsf{fma}\left(1 - ux, \left(zi \cdot ux\right) \cdot maxCos, \sqrt{\mathsf{fma}\left(\left(ux - 1\right) \cdot maxCos, \left(\color{blue}{maxCos} \cdot ux\right) \cdot ux, 1\right)} \cdot \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 20: 81.8% accurate, 6.1× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ xi (fma 2.0 (* uy (* yi PI)) (* maxCos (* ux (* zi (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + fmaf(2.0f, (uy * (yi * ((float) M_PI))), (maxCos * (ux * (zi * (1.0f - ux)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + fma(Float32(2.0), Float32(uy * Float32(yi * Float32(pi))), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

7. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto xi + \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   5. lower-PI.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   9. lower--.f32 81.8%

      \[\leadsto xi + \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

10. Applied rewrites 81.8%

    \[\leadsto xi + \color{blue}{\mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 21: 51.6% accurate, 10.5× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ xi (* maxCos (* ux (* zi (- 1.0 ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (maxCos * (ux * (zi * (1.0f - ux))));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (maxcos * (ux * (zi * (1.0e0 - ux))))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (maxCos * (ux * (zi * (single(1.0) - ux))));
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.4%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right)\right) \]

   5. lower--.f32 51.6%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \]

10. Applied rewrites 51.6%

    \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
11. Add Preprocessing

Alternative 22: 49.4% accurate, 16.6× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[xi + \left(zi \cdot maxCos\right) \cdot ux \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ xi (* (* zi maxCos) ux)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + ((zi * maxCos) * ux);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + ((zi * maxcos) * ux)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(Float32(zi * maxCos) * ux))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + ((zi * maxCos) * ux);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.4%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   2. *-commutative N/A

      \[\leadsto xi + \left(ux \cdot zi\right) \cdot maxCos \]

   3. lift-*.f32 N/A

      \[\leadsto xi + \left(ux \cdot zi\right) \cdot maxCos \]

   4. *-commutative N/A

      \[\leadsto xi + \left(zi \cdot ux\right) \cdot maxCos \]

   5. associate-*l* N/A

      \[\leadsto xi + zi \cdot \left(ux \cdot \color{blue}{maxCos}\right) \]

   6. *-commutative N/A

      \[\leadsto xi + zi \cdot \left(maxCos \cdot ux\right) \]

   7. associate-*r* N/A

      \[\leadsto xi + \left(zi \cdot maxCos\right) \cdot ux \]

   8. lower-*.f32 N/A

      \[\leadsto xi + \left(zi \cdot maxCos\right) \cdot ux \]

   9. lower-*.f32 49.4%

      \[\leadsto xi + \left(zi \cdot maxCos\right) \cdot ux \]

9. Applied rewrites 49.4%

   \[\leadsto xi + \left(zi \cdot maxCos\right) \cdot ux \]
10. Add Preprocessing

Alternative 23: 49.4% accurate, 17.7× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma (* zi maxCos) ux xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((zi * maxCos), ux, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(zi * maxCos), ux, xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.4%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

   4. associate-*l* N/A

      \[\leadsto xi + \left(maxCos \cdot ux\right) \cdot zi \]

   5. lift-*.f32 N/A

      \[\leadsto xi + \left(maxCos \cdot ux\right) \cdot zi \]

   6. lift-*.f32 N/A

      \[\leadsto xi + \left(maxCos \cdot ux\right) \cdot zi \]

   7. +-commutative N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   8. lift-*.f32 N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   9. *-commutative N/A

      \[\leadsto zi \cdot \left(maxCos \cdot ux\right) + xi \]

   10. lift-*.f32 N/A

       \[\leadsto zi \cdot \left(maxCos \cdot ux\right) + xi \]

   11. associate-*r* N/A

       \[\leadsto \left(zi \cdot maxCos\right) \cdot ux + xi \]

   12. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]

   13. lower-*.f32 49.4%

       \[\leadsto \mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]

9. Applied rewrites 49.4%

   \[\leadsto \mathsf{fma}\left(zi \cdot maxCos, ux, xi\right) \]
10. Add Preprocessing

Alternative 24: 49.4% accurate, 17.7× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (fma (* maxCos ux) zi xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((maxCos * ux), zi, xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(maxCos * ux), zi, xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.4%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

   4. associate-*l* N/A

      \[\leadsto xi + \left(maxCos \cdot ux\right) \cdot zi \]

   5. lift-*.f32 N/A

      \[\leadsto xi + \left(maxCos \cdot ux\right) \cdot zi \]

   6. lift-*.f32 N/A

      \[\leadsto xi + \left(maxCos \cdot ux\right) \cdot zi \]

   7. +-commutative N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   8. lift-*.f32 N/A

      \[\leadsto \left(maxCos \cdot ux\right) \cdot zi + xi \]

   9. lower-fma.f32 49.4%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]

9. Applied rewrites 49.4%

   \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi, xi\right) \]
10. Add Preprocessing

Alternative 25: 12.1% accurate, 22.8× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(maxCos \cdot zi\right) \cdot ux \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (* maxCos zi) ux))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (maxCos * zi) * ux;
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (maxcos * zi) * ux
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(maxCos * zi) * ux)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (maxCos * zi) * ux;
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.4%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 12.1%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.1%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

    2. lift-*.f32 N/A

       \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

    3. *-commutative N/A

       \[\leadsto maxCos \cdot \left(zi \cdot ux\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]

    5. lower-*.f32 N/A

       \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]

    6. lower-*.f32 12.1%

       \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]

12. Applied rewrites 12.1%

    \[\leadsto \left(maxCos \cdot zi\right) \cdot ux \]
13. Add Preprocessing

Alternative 26: 12.1% accurate, 22.8× speedup

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[maxCos \cdot \left(ux \cdot zi\right) \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0))
                         (and (<= -10000.0 yi) (<= yi 10000.0)))
                    (and (<= -10000.0 zi) (<= zi 10000.0)))
               (and (<= 2.328306437e-10 ux) (<= ux 1.0)))
          (and (<= 2.328306437e-10 uy) (<= uy 1.0)))
     (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   4. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   6. lower-sqrt.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

   7. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right) \]

4. Applied rewrites 51.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto xi + maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]

   3. lower-*.f32 49.4%

      \[\leadsto xi + maxCos \cdot \left(ux \cdot zi\right) \]

7. Applied rewrites 49.4%

   \[\leadsto xi + \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

8. Taylor expanded in xi around 0

   \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

   2. lower-*.f32 12.1%

      \[\leadsto maxCos \cdot \left(ux \cdot zi\right) \]

10. Applied rewrites 12.1%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025356 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))