UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 11.4s

Alternatives: 20

Speedup: 0.8×

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: 98.9% 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: 98.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (* (* uy 2.0) PI))
        (t_2 (* (* ux ux) (* ux ux))))
   (+
    (+
     (* (* (cos t_1) (sqrt (- 1.0 (* t_0 t_0)))) xi)
     (*
      (*
       (sin t_1)
       (sqrt
        (*
         t_2
         (-
          (/ 1.0 t_2)
          (fma
           -2.0
           (/ (* maxCos maxCos) ux)
           (fma maxCos maxCos (/ (* maxCos maxCos) (* ux ux))))))))
      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 = (uy * 2.0f) * ((float) M_PI);
	float t_2 = (ux * ux) * (ux * ux);
	return (((cosf(t_1) * sqrtf((1.0f - (t_0 * t_0)))) * xi) + ((sinf(t_1) * sqrtf((t_2 * ((1.0f / t_2) - fmaf(-2.0f, ((maxCos * maxCos) / ux), fmaf(maxCos, maxCos, ((maxCos * maxCos) / (ux * ux)))))))) * 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(uy * Float32(2.0)) * Float32(pi))
	t_2 = Float32(Float32(ux * ux) * Float32(ux * ux))
	return Float32(Float32(Float32(Float32(cos(t_1) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) * xi) + Float32(Float32(sin(t_1) * sqrt(Float32(t_2 * Float32(Float32(Float32(1.0) / t_2) - fma(Float32(-2.0), Float32(Float32(maxCos * maxCos) / ux), fma(maxCos, maxCos, Float32(Float32(maxCos * maxCos) / Float32(ux * ux)))))))) * yi)) + Float32(t_0 * zi))
end

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{{ux}^{4} \cdot \color{blue}{\left(\frac{1}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. sqr-pow N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{\left(\frac{4}{2}\right)} \cdot {ux}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\color{blue}{\frac{1}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{2} \cdot {ux}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\frac{1}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{2} \cdot {ux}^{2}\right) \cdot \left(\frac{1}{{ux}^{\color{blue}{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{2} \cdot {ux}^{2}\right) \cdot \left(\color{blue}{\frac{1}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. unpow2 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot {ux}^{2}\right) \cdot \left(\frac{\color{blue}{1}}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot {ux}^{2}\right) \cdot \left(\frac{\color{blue}{1}}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(\frac{1}{\color{blue}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(\frac{1}{\color{blue}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   10. lower--.f32 N/A

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(\frac{1}{{ux}^{4}} - \color{blue}{\left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied rewrites 98.9%

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

Alternative 2: 98.9% 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

Derivation

1. Initial program 98.9%

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

Alternative 3: 98.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)))
   (+
    (+
     (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0)))) xi)
     (* (sin (* (+ uy uy) PI)) 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;
	return (((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) * xi) + (sinf(((uy + uy) * ((float) M_PI))) * yi)) + (t_0 * zi);
}

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. lower-sin.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. lower-+.f32 N/A

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

   6. lift-PI.f32 98.8

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

4. Applied rewrites 98.8%

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

Alternative 4: 98.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ uy uy) PI)))
   (+
    (fma yi (sin t_0) (* xi (cos t_0)))
    (* (* (* (- 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 + uy) * ((float) M_PI);
	return fmaf(yi, sinf(t_0), (xi * cosf(t_0))) + ((((1.0f - ux) * maxCos) * ux) * zi);
}

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-cos.f32 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 N/A

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

   15. lift-PI.f32 98.7

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

4. Applied rewrites 98.7%

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

Alternative 5: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 98.7%

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

Alternative 6: 96.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{{ux}^{4} \cdot \color{blue}{\left(\frac{1}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. sqr-pow N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{\left(\frac{4}{2}\right)} \cdot {ux}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\color{blue}{\frac{1}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{2} \cdot {ux}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\frac{1}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{2} \cdot {ux}^{2}\right) \cdot \left(\frac{1}{{ux}^{\color{blue}{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left({ux}^{2} \cdot {ux}^{2}\right) \cdot \left(\color{blue}{\frac{1}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. unpow2 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot {ux}^{2}\right) \cdot \left(\frac{\color{blue}{1}}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot {ux}^{2}\right) \cdot \left(\frac{\color{blue}{1}}{{ux}^{4}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(\frac{1}{\color{blue}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(\frac{1}{\color{blue}{{ux}^{4}}} - \left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   10. lower--.f32 N/A

       \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(\frac{1}{{ux}^{4}} - \color{blue}{\left(-2 \cdot \frac{{maxCos}^{2}}{ux} + \left(\frac{{maxCos}^{2}}{{ux}^{2}} + {maxCos}^{2}\right)\right)}\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied rewrites 98.9%

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

5. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-cos.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-*.f32 N/A

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

7. Applied rewrites 95.9%

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

Alternative 7: 96.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

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

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

   5. lower-fma.f32 N/A

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

4. Applied rewrites 95.9%

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

Alternative 8: 95.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00034500000765547156:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right), xi \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(\left(uy + uy\right) \cdot \pi\right), xi \cdot \sin \left(\mathsf{fma}\left(uy + uy, \pi, \frac{\pi}{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00034500000765547156)
   (fma
    (* maxCos ux)
    (* zi (- 1.0 ux))
    (fma
     uy
     (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))
     (*
      xi
      (sqrt
       (-
        1.0
        (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux)))))))))
   (fma yi (sin (* (+ uy uy) PI)) (* xi (sin (fma (+ uy uy) PI (/ PI 2.0)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00034500000765547156f) {
		tmp = fmaf((maxCos * ux), (zi * (1.0f - ux)), fmaf(uy, fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI)))), (xi * sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))))));
	} else {
		tmp = fmaf(yi, sinf(((uy + uy) * ((float) M_PI))), (xi * sinf(fmaf((uy + uy), ((float) M_PI), (((float) M_PI) / 2.0f)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00034500000765547156))
		tmp = fma(Float32(maxCos * ux), Float32(zi * Float32(Float32(1.0) - ux)), fma(uy, fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))), Float32(xi * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))))));
	else
		tmp = fma(yi, sin(Float32(Float32(uy + uy) * Float32(pi))), Float32(xi * sin(fma(Float32(uy + uy), Float32(pi), Float32(Float32(pi) / Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.45000008e-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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 99.2

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

   7. Applied rewrites 99.2%

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

   if 3.45000008e-4 < uy

   1. Initial program 98.2%

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

   2. Taylor expanded in 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-cos.f32 N/A

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

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

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

      3. lower-sin.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lift-PI.f32 91.0

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

   6. Applied rewrites 91.0%

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

Alternative 9: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy + uy\right) \cdot \pi\\ \mathbf{if}\;uy \leq 0.00034500000765547156:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right), xi \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ uy uy) PI)))
   (if (<= uy 0.00034500000765547156)
     (fma
      (* maxCos ux)
      (* zi (- 1.0 ux))
      (fma
       uy
       (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))
       (*
        xi
        (sqrt
         (-
          1.0
          (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux)))))))))
     (fma yi (sin t_0) (* xi (cos t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (uy + uy) * ((float) M_PI);
	float tmp;
	if (uy <= 0.00034500000765547156f) {
		tmp = fmaf((maxCos * ux), (zi * (1.0f - ux)), fmaf(uy, fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI)))), (xi * sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))))));
	} else {
		tmp = fmaf(yi, sinf(t_0), (xi * cosf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy + uy) * Float32(pi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.00034500000765547156))
		tmp = fma(Float32(maxCos * ux), Float32(zi * Float32(Float32(1.0) - ux)), fma(uy, fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))), Float32(xi * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))))));
	else
		tmp = fma(yi, sin(t_0), Float32(xi * cos(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.45000008e-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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 99.2

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

   7. Applied rewrites 99.2%

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

   if 3.45000008e-4 < uy

   1. Initial program 98.2%

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

   2. Taylor expanded in 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.2%

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

Alternative 10: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00034500000765547156:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right), xi \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00034500000765547156)
   (fma
    (* maxCos ux)
    (* zi (- 1.0 ux))
    (fma
     uy
     (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))
     (*
      xi
      (sqrt
       (-
        1.0
        (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux)))))))))
   (fma
    yi
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (* xi (cos (* (+ uy uy) PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00034500000765547156f) {
		tmp = fmaf((maxCos * ux), (zi * (1.0f - ux)), fmaf(uy, fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI)))), (xi * sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))))));
	} else {
		tmp = fmaf(yi, (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))), (xi * cosf(((uy + uy) * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00034500000765547156))
		tmp = fma(Float32(maxCos * ux), Float32(zi * Float32(Float32(1.0) - ux)), fma(uy, fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))), Float32(xi * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))))));
	else
		tmp = fma(yi, Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))), Float32(xi * cos(Float32(Float32(uy + uy) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.45000008e-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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 99.2

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

   7. Applied rewrites 99.2%

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

   if 3.45000008e-4 < uy

   1. Initial program 98.2%

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

   2. Taylor expanded in 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(yi, uy \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}, xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      7. pow2 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      11. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      12. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

      15. lift-PI.f32 78.1

          \[\leadsto \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto \mathsf{fma}\left(yi, uy \cdot \color{blue}{\mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)}, xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00034500000765547156:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right), xi \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(\left(uy + uy\right) \cdot \pi\right), xi \cdot \left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00034500000765547156)
   (fma
    (* maxCos ux)
    (* zi (- 1.0 ux))
    (fma
     uy
     (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))
     (*
      xi
      (sqrt
       (-
        1.0
        (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux)))))))))
   (fma
    yi
    (sin (* (+ uy uy) PI))
    (* xi (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00034500000765547156f) {
		tmp = fmaf((maxCos * ux), (zi * (1.0f - ux)), fmaf(uy, fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI)))), (xi * sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))))));
	} else {
		tmp = fmaf(yi, sinf(((uy + uy) * ((float) M_PI))), (xi * (1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00034500000765547156))
		tmp = fma(Float32(maxCos * ux), Float32(zi * Float32(Float32(1.0) - ux)), fma(uy, fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))), Float32(xi * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))))));
	else
		tmp = fma(yi, sin(Float32(Float32(uy + uy) * Float32(pi))), Float32(xi * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.45000008e-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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 99.2

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

   7. Applied rewrites 99.2%

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

   if 3.45000008e-4 < uy

   1. Initial program 98.2%

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

   2. Taylor expanded in 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 76.8

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

   7. Applied rewrites 76.8%

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

Alternative 12: 90.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00034500000765547156:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(\left(uy + uy\right) \cdot \pi\right), xi \cdot \left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00034500000765547156)
   (+
    xi
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
   (fma
    yi
    (sin (* (+ uy uy) PI))
    (* xi (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00034500000765547156f) {
		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(yi, sinf(((uy + uy) * ((float) M_PI))), (xi * (1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00034500000765547156))
		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(yi, sin(Float32(Float32(uy + uy) * Float32(pi))), Float32(xi * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.45000008e-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

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

   4. Applied rewrites 99.3%

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

   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 99.0%

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

   if 3.45000008e-4 < uy

   1. Initial program 98.2%

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

   2. Taylor expanded in 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 76.8

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

   7. Applied rewrites 76.8%

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

Alternative 13: 88.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.035999998450279236:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, 2 \cdot \left(uy \cdot \pi\right), xi \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.035999998450279236)
   (+
    xi
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
   (fma yi (* 2.0 (* uy PI)) (* xi (cos (* (+ uy uy) PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.035999998450279236f) {
		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(yi, (2.0f * (uy * ((float) M_PI))), (xi * cosf(((uy + uy) * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.035999998450279236))
		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(yi, Float32(Float32(2.0) * Float32(uy * Float32(pi))), Float32(xi * cos(Float32(Float32(uy + uy) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0359999985

   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

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

   4. Applied rewrites 94.9%

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

   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 94.6%

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

   if 0.0359999985 < uy

   1. Initial program 97.3%

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

   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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 54.5

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

   7. Applied rewrites 54.5%

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

Alternative 14: 85.5% 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 98.9%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 85.7%

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

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 85.5%

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

Alternative 15: 82.9% 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 98.9%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 85.7%

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

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 82.9%

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

Alternative 16: 81.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 85.7%

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

5. Taylor expanded in uy around 0

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

   1. pow2 N/A

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

   2. pow2 N/A

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

   3. pow2 N/A

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

7. Applied rewrites 81.5%

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

8. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 81.4

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

10. Applied rewrites 81.4%

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

Alternative 17: 73.9% 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 98.9%

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

2. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-cos.f32 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

4. Applied rewrites 90.5%

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

5. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   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 N/A

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

   5. lift-*.f32 73.9

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

7. Applied rewrites 73.9%

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

Alternative 18: 55.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;yi \leq -3.0000001167615996 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;xi\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= yi -3.0000001167615996e-17)
   (* 2.0 (* uy (* yi PI)))
   (if (<= yi 4.99999991225835e-14) xi (* yi (* 2.0 (* uy PI))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (yi <= -3.0000001167615996e-17f) {
		tmp = 2.0f * (uy * (yi * ((float) M_PI)));
	} else if (yi <= 4.99999991225835e-14f) {
		tmp = xi;
	} else {
		tmp = yi * (2.0f * (uy * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (yi <= Float32(-3.0000001167615996e-17))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))));
	elseif (yi <= Float32(4.99999991225835e-14))
		tmp = xi;
	else
		tmp = Float32(yi * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (yi <= single(-3.0000001167615996e-17))
		tmp = single(2.0) * (uy * (yi * single(pi)));
	elseif (yi <= single(4.99999991225835e-14))
		tmp = xi;
	else
		tmp = yi * (single(2.0) * (uy * single(pi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if yi < -3.0000001e-17

   1. Initial program 98.7%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in xi around 0

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

      1. lower-fma.f32 N/A

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

   7. Applied rewrites 51.2%

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

   8. Taylor expanded in ux around 0

      \[\leadsto 2 \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right) \]
   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. lift-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 48.9

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

   10. Applied rewrites 48.9%

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

   if -3.0000001e-17 < yi < 4.99999991e-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 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in uy around 0

      \[\leadsto xi \]
   6. Step-by-step derivation

   7. Applied rewrites 59.7%

      \[\leadsto xi \]

   if 4.99999991e-14 < yi

   1. Initial program 98.7%

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

   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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in xi around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sin.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in uy around 0

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 51.9

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

   10. Applied rewrites 51.9%

       \[\leadsto yi \cdot \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 55.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := yi \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{if}\;yi \leq -3.0000001167615996 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;xi\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* yi (* 2.0 (* uy PI)))))
   (if (<= yi -3.0000001167615996e-17)
     t_0
     (if (<= yi 4.99999991225835e-14) xi t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = yi * (2.0f * (uy * ((float) M_PI)));
	float tmp;
	if (yi <= -3.0000001167615996e-17f) {
		tmp = t_0;
	} else if (yi <= 4.99999991225835e-14f) {
		tmp = xi;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(yi * Float32(Float32(2.0) * Float32(uy * Float32(pi))))
	tmp = Float32(0.0)
	if (yi <= Float32(-3.0000001167615996e-17))
		tmp = t_0;
	elseif (yi <= Float32(4.99999991225835e-14))
		tmp = xi;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = yi * (single(2.0) * (uy * single(pi)));
	tmp = single(0.0);
	if (yi <= single(-3.0000001167615996e-17))
		tmp = t_0;
	elseif (yi <= single(4.99999991225835e-14))
		tmp = xi;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.0000001e-17 or 4.99999991e-14 < yi

   1. Initial program 98.7%

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

   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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in xi around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sin.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 61.6

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

   7. Applied rewrites 61.6%

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

   8. Taylor expanded in uy around 0

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 50.2

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

   10. Applied rewrites 50.2%

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

   if -3.0000001e-17 < yi < 4.99999991e-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 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

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in uy around 0

      \[\leadsto xi \]
   6. Step-by-step derivation

   7. Applied rewrites 59.7%

      \[\leadsto xi \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 45.5% accurate, 155.4× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return 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 = xi
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in 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

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

   2. lower-fma.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-cos.f32 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

4. Applied rewrites 90.5%

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

5. Taylor expanded in uy around 0

   \[\leadsto xi \]
6. Step-by-step derivation

7. Applied rewrites 45.5%

   \[\leadsto xi \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(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)))