UniformSampleCone 2

Percentage Accurate: 99.0% → 99.0%

Time: 17.1s

Alternatives: 22

Speedup: 1.0×

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

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (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

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* PI (+ uy uy)))
        (t_2 (sqrt (- 1.0 (* t_0 t_0)))))
   (fma (* (cos t_1) t_2) xi (fma (sin t_1) (* t_2 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 = ((float) M_PI) * (uy + uy);
	float t_2 = sqrtf((1.0f - (t_0 * t_0)));
	return fmaf((cosf(t_1) * t_2), xi, fmaf(sinf(t_1), (t_2 * 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 = Float32(Float32(pi) * Float32(uy + uy))
	t_2 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	return fma(Float32(cos(t_1) * t_2), xi, fma(sin(t_1), Float32(t_2 * yi), Float32(t_0 * zi)))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 99.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot ux\right)\\ t_1 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_1 \cdot t\_0\right) \cdot xi + \left(\sin t\_1 \cdot t\_0\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (+
          1.0
          (* (fma -0.5 (* maxCos maxCos) (* (* maxCos maxCos) ux)) (* ux ux))))
        (t_1 (* (* uy 2.0) PI)))
   (+
    (+ (* (* (cos t_1) t_0) xi) (* (* (sin t_1) t_0) yi))
    (* (* (* (- 1.0 ux) maxCos) ux) zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 1.0f + (fmaf(-0.5f, (maxCos * maxCos), ((maxCos * maxCos) * ux)) * (ux * ux));
	float t_1 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_1) * t_0) * xi) + ((sinf(t_1) * t_0) * yi)) + ((((1.0f - ux) * maxCos) * ux) * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) + Float32(fma(Float32(-0.5), Float32(maxCos * maxCos), Float32(Float32(maxCos * maxCos) * ux)) * Float32(ux * ux)))
	t_1 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_1) * t_0) * xi) + Float32(Float32(sin(t_1) * t_0) * yi)) + Float32(Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux) * zi))
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.9

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot \color{blue}{ux}\right)\right)\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied rewrites 98.9%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\left(1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot ux\right)\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.8

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot ux\right)\right)\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot \color{blue}{ux}\right)\right)\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot ux\right)\right)\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\left(1 + \mathsf{fma}\left(-0.5, maxCos \cdot maxCos, \left(maxCos \cdot maxCos\right) \cdot ux\right) \cdot \left(ux \cdot ux\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
8. Add Preprocessing

Alternative 3: 98.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy 2.0) PI)))
   (+
    (+ (* (* (cos t_0) 1.0) xi) (* (* (sin t_0) 1.0) yi))
    (* (* (* (- 1.0 ux) maxCos) ux) zi))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 98.8%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot 1\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{1}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
8. Add Preprocessing

Alternative 4: 96.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;uy \leq 0.0006500000017695129:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (if (<= uy 0.0006500000017695129)
     (+
      xi
      (fma
       maxCos
       (* ux (* zi (- 1.0 ux)))
       (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
     (fma (cos t_0) xi (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (uy <= 0.0006500000017695129f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = fmaf(cosf(t_0), xi, (sinf(t_0) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0006500000017695129))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(cos(t_0), xi, Float32(sin(t_0) * yi));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 6.50000002e-4

   1. Initial program 99.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 98.8%

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

   if 6.50000002e-4 < uy

   1. Initial program 98.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 90.5%

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

Alternative 5: 95.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (fma (* maxCos ux) zi (fma (cos t_0) xi (* (sin t_0) yi)))))

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((maxCos * ux), zi, fmaf(cosf(t_0), xi, (sinf(t_0) * yi)));
}

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

4. Applied rewrites 95.9%

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

Alternative 6: 90.2% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (*
  (+
   (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
   (/ (fma maxCos (* ux zi) (* yi (sin (* 2.0 (* uy PI))))) xi))
  xi))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in xi around inf

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f32 N/A

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-PI.f32 90.2

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

10. Applied rewrites 90.2%

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

Alternative 7: 89.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.007000000216066837:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}{xi}\right) \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.007000000216066837)
   (+
    xi
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
   (*
    (+ 1.0 (/ (fma maxCos (* ux zi) (* yi (sin (* 2.0 (* uy PI))))) xi))
    xi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.007000000216066837f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = (1.0f + (fmaf(maxCos, (ux * zi), (yi * sinf((2.0f * (uy * ((float) M_PI)))))) / xi)) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.007000000216066837))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = Float32(Float32(Float32(1.0) + Float32(fma(maxCos, Float32(ux * zi), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))) / xi)) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00700000022

   1. Initial program 99.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 97.3%

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

   if 0.00700000022 < uy

   1. Initial program 97.9%

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

   2. Taylor expanded in xi around inf

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f32 N/A

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

   7. Applied rewrites 94.6%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 61.4%

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

Alternative 8: 87.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.18000000715255737:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, \mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot uy, \left(xi \cdot \left(\pi \cdot \pi\right)\right) \cdot 1, \left(2 \cdot yi\right) \cdot \left(1 \cdot \pi\right)\right), uy, 1 \cdot xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)}{xi} \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.18000000715255737)
   (fma
    (* maxCos ux)
    (* (- 1.0 ux) zi)
    (fma
     (fma (* -2.0 uy) (* (* xi (* PI PI)) 1.0) (* (* 2.0 yi) (* 1.0 PI)))
     uy
     (* 1.0 xi)))
   (* (/ (fma maxCos (* ux zi) (* yi (sin (* 2.0 (* uy PI))))) xi) xi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.18000000715255737f) {
		tmp = fmaf((maxCos * ux), ((1.0f - ux) * zi), fmaf(fmaf((-2.0f * uy), ((xi * (((float) M_PI) * ((float) M_PI))) * 1.0f), ((2.0f * yi) * (1.0f * ((float) M_PI)))), uy, (1.0f * xi)));
	} else {
		tmp = (fmaf(maxCos, (ux * zi), (yi * sinf((2.0f * (uy * ((float) M_PI)))))) / xi) * xi;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.18000000715255737))
		tmp = fma(Float32(maxCos * ux), Float32(Float32(Float32(1.0) - ux) * zi), fma(fma(Float32(Float32(-2.0) * uy), Float32(Float32(xi * Float32(Float32(pi) * Float32(pi))) * Float32(1.0)), Float32(Float32(Float32(2.0) * yi) * Float32(Float32(1.0) * Float32(pi)))), uy, Float32(Float32(1.0) * xi)));
	else
		tmp = Float32(Float32(fma(maxCos, Float32(ux * zi), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))) / xi) * xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.180000007

   1. Initial program 99.2%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 91.4%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 91.3%

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

   if 0.180000007 < uy

   1. Initial program 96.4%

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

   2. Taylor expanded in xi around inf

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f32 N/A

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in xi around 0

      \[\leadsto \frac{maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{xi} \cdot xi \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

         \[\leadsto \frac{maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi} \cdot xi \]

      3. lift-*.f32 N/A

         \[\leadsto \frac{maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi} \cdot xi \]

      4. lift-sin.f32 N/A

         \[\leadsto \frac{maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi} \cdot xi \]

      5. lift-*.f32 N/A

         \[\leadsto \frac{maxCos \cdot \left(ux \cdot zi\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi} \cdot xi \]

      6. lift-fma.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-/.f32 47.8

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

   10. Applied rewrites 47.8%

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

Alternative 9: 87.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \left(\cos t\_0 + \frac{\mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot t\_0\right)}{xi}\right) \cdot xi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (* (+ (cos t_0) (/ (fma maxCos (* ux zi) (* yi t_0)) xi)) xi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return (cosf(t_0) + (fmaf(maxCos, (ux * zi), (yi * t_0)) / xi)) * xi;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return Float32(Float32(cos(t_0) + Float32(fma(maxCos, Float32(ux * zi), Float32(yi * t_0)) / xi)) * xi)
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in xi around inf

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f32 N/A

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in uy around 0

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 87.3

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

10. Applied rewrites 87.3%

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

Alternative 10: 86.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (* maxCos ux)
  (* (- 1.0 ux) zi)
  (fma
   (fma (* -2.0 uy) (* (* xi (* PI PI)) 1.0) (* (* 2.0 yi) (* 1.0 PI)))
   uy
   (* 1.0 xi))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 86.1%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 86.1%

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

11. Taylor expanded in ux around 0

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

13. Applied rewrites 86.0%

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

Alternative 11: 86.0% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (fma
   maxCos
   (* ux (* zi (- 1.0 ux)))
   (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

7. Applied rewrites 86.0%

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

Alternative 12: 83.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ xi + \mathsf{fma}\left(maxCos, ux \cdot zi, uy \cdot \left(uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \frac{yi \cdot \pi}{uy}\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (fma
   maxCos
   (* ux zi)
   (* uy (* uy (fma -2.0 (* xi (* PI PI)) (* 2.0 (/ (* yi PI) uy))))))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.3%

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

8. Taylor expanded in uy around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. pow2 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 83.3

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

10. Applied rewrites 83.3%

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

Alternative 13: 83.3% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (fma
   maxCos
   (* ux zi)
   (fma -2.0 (* (* uy uy) (* xi (* PI PI))) (* 2.0 (* uy (* yi PI)))))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.3%

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

8. Taylor expanded in xi around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 83.3

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

10. Applied rewrites 83.3%

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

Alternative 14: 83.3% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  xi
  (fma
   maxCos
   (* ux zi)
   (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.3%

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

Alternative 15: 81.9% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (fma (+ uy uy) (* (* yi PI) 1.0) (* 1.0 xi))
  (* (* (* (- 1.0 ux) maxCos) ux) zi)))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 81.9%

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

Alternative 16: 81.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(xi - -2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ (- xi (* -2.0 (* uy (* yi PI)))) (* (* (* (- 1.0 ux) maxCos) ux) zi)))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (xi - (single(-2.0) * (uy * (yi * single(pi))))) + ((((single(1.0) - ux) * maxCos) * ux) * zi);
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 81.9

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

7. Applied rewrites 81.9%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. lower--.f32 N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 81.9

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

9. Applied rewrites 81.9%

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

Alternative 17: 79.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right) + \left(maxCos \cdot ux\right) \cdot zi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ (+ xi (* 2.0 (* uy (* yi PI)))) (* (* maxCos ux) zi)))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (xi + (single(2.0) * (uy * (yi * single(pi))))) + ((maxCos * ux) * zi);
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 81.9

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

7. Applied rewrites 81.9%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 79.3%

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

Alternative 18: 79.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ xi + \mathsf{fma}\left(maxCos, ux \cdot zi, uy \cdot \left(2 \cdot \left(yi \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (fma maxCos (* ux zi) (* uy (* 2.0 (* yi PI))))))

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.3%

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

8. Taylor expanded in xi around 0

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 79.3

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

10. Applied rewrites 79.3%

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

Alternative 19: 74.5% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (* 2.0 (* uy (* yi PI)))))

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (single(2.0) * (uy * (yi * single(pi))));
end

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.3%

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

8. Taylor expanded in yi around inf

   \[\leadsto xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. lift-*.f32 N/A

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. lift-PI.f32 74.5

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

10. Applied rewrites 74.5%

    \[\leadsto xi + 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \pi\right)}\right) \]
11. Add Preprocessing

Alternative 20: 59.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\\ \mathbf{if}\;yi \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 5.500000072890775 \cdot 10^{-14}:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot zi\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* yi PI)))))
   (if (<= yi -9.999999717180685e-10)
     t_0
     (if (<= yi 5.500000072890775e-14) (+ xi (* maxCos (* ux zi))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * (yi * ((float) M_PI)));
	float tmp;
	if (yi <= -9.999999717180685e-10f) {
		tmp = t_0;
	} else if (yi <= 5.500000072890775e-14f) {
		tmp = xi + (maxCos * (ux * zi));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))
	tmp = Float32(0.0)
	if (yi <= Float32(-9.999999717180685e-10))
		tmp = t_0;
	elseif (yi <= Float32(5.500000072890775e-14))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * zi)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * (yi * single(pi)));
	tmp = single(0.0);
	if (yi <= single(-9.999999717180685e-10))
		tmp = t_0;
	elseif (yi <= single(5.500000072890775e-14))
		tmp = xi + (maxCos * (ux * zi));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -9.99999972e-10 or 5.50000007e-14 < yi

   1. Initial program 98.8%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in yi around inf

      \[\leadsto 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      2. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lift-*.f32 N/A

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. lift-PI.f32 54.1

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

   10. Applied rewrites 54.1%

       \[\leadsto 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \pi\right)}\right) \]

   if -9.99999972e-10 < yi < 5.50000007e-14

   1. Initial program 99.1%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in zi around inf

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

      1. lower-*.f32 N/A

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

      2. lift-*.f32 62.8

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

   10. Applied rewrites 62.8%

       \[\leadsto xi + maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.0% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ xi + maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (* maxCos (* ux zi))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.1%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.3%

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

8. Taylor expanded in zi around inf

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

   1. lower-*.f32 N/A

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

   2. lift-*.f32 50.0

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

10. Applied rewrites 50.0%

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

Alternative 22: 46.0% accurate, 39.1× speedup

Math

\[\begin{array}{l} \\ 1 \cdot xi \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* 1.0 xi))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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 = 1.0e0 * xi
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in xi around inf

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. div-add-rev N/A

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

   7. lower-/.f32 N/A

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

7. Applied rewrites 95.7%

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

8. Taylor expanded in xi around inf

   \[\leadsto \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot xi \]

   2. lift-PI.f32 N/A

      \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi \]

   3. lift-*.f32 N/A

      \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi \]

   4. lift-cos.f32 53.2

      \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi \]

10. Applied rewrites 53.2%

    \[\leadsto \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi \]

11. Taylor expanded in uy around 0

    \[\leadsto 1 \cdot xi \]
12. Step-by-step derivation

13. Applied rewrites 46.0%

    \[\leadsto 1 \cdot xi \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025133 
(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)))