UniformSampleCone, y

Percentage Accurate: 58.1% → 98.1%

Time: 7.5s

Alternatives: 19

Speedup: 1.2×

Specification

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

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

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

Julia

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

MATLAB

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

Initial Program: 58.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

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

Julia

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

MATLAB

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

Alternative 1: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2}{ux}, 1 + \frac{2}{ux} \cdot 1\right)} \cdot ux\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt
   (*
    (fma
     (fma (- maxCos) ux (* (- 2.0 (/ 2.0 ux)) ux))
     maxCos
     (*
      (/
       (- (pow (/ 2.0 ux) 3.0) 1.0)
       (fma (/ 2.0 ux) (/ 2.0 ux) (+ 1.0 (* (/ 2.0 ux) 1.0))))
      ux))
    ux))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((fmaf(fmaf(-maxCos, ux, ((2.0f - (2.0f / ux)) * ux)), maxCos, (((powf((2.0f / ux), 3.0f) - 1.0f) / fmaf((2.0f / ux), (2.0f / ux), (1.0f + ((2.0f / ux) * 1.0f)))) * ux)) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(fma(fma(Float32(-maxCos), ux, Float32(Float32(Float32(2.0) - Float32(Float32(2.0) / ux)) * ux)), maxCos, Float32(Float32(Float32((Float32(Float32(2.0) / ux) ^ Float32(3.0)) - Float32(1.0)) / fma(Float32(Float32(2.0) / ux), Float32(Float32(2.0) / ux), Float32(Float32(1.0) + Float32(Float32(Float32(2.0) / ux) * Float32(1.0))))) * ux)) * ux)))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in ux around inf

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(\left(-1 \cdot {\left(maxCos - 1\right)}^{2} + 2 \cdot \frac{1}{ux}\right) - 2 \cdot \frac{maxCos}{ux}\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 98.2%

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

9. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(maxCos \cdot \left(-1 \cdot \left(maxCos \cdot ux\right) + ux \cdot \left(2 - 2 \cdot \frac{1}{ux}\right)\right) + ux \cdot \left(2 \cdot \frac{1}{ux} - 1\right)\right) \cdot ux} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. associate-*r* N/A

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

    4. mul-1-neg N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-neg.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. lower--.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 N/A

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

    13. *-commutative N/A

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

11. Applied rewrites 98.2%

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

    1. lift--.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. metadata-eval N/A

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

    4. associate-*r/ N/A

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

    5. flip3-- N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(2 \cdot \frac{1}{ux}\right)}^{3} - {1}^{3}}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    6. lower-/.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(2 \cdot \frac{1}{ux}\right)}^{3} - {1}^{3}}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    7. metadata-eval N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(2 \cdot \frac{1}{ux}\right)}^{3} - 1}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    8. lower--.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(2 \cdot \frac{1}{ux}\right)}^{3} - 1}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    9. lower-pow.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(2 \cdot \frac{1}{ux}\right)}^{3} - 1}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    10. associate-*r/ N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2 \cdot 1}{ux}\right)}^{3} - 1}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    11. metadata-eval N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    12. lift-/.f32 N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(2 \cdot \frac{1}{ux}\right) + \left(1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    13. lower-fma.f32 N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(2 \cdot \frac{1}{ux}, 2 \cdot \frac{1}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    14. associate-*r/ N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2 \cdot 1}{ux}, 2 \cdot \frac{1}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    15. metadata-eval N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, 2 \cdot \frac{1}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    16. lift-/.f32 N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, 2 \cdot \frac{1}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    17. associate-*r/ N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2 \cdot 1}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    18. metadata-eval N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    19. lift-/.f32 N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2}{ux}, 1 \cdot 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    20. metadata-eval N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2}{ux}, 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

    21. lower-+.f32 N/A

        \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2}{ux}, 1 + \left(2 \cdot \frac{1}{ux}\right) \cdot 1\right)} \cdot ux\right) \cdot ux} \]

13. Applied rewrites 98.1%

    \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-maxCos, ux, \left(2 - \frac{2}{ux}\right) \cdot ux\right), maxCos, \frac{{\left(\frac{2}{ux}\right)}^{3} - 1}{\mathsf{fma}\left(\frac{2}{ux}, \frac{2}{ux}, 1 + \frac{2}{ux} \cdot 1\right)} \cdot ux\right) \cdot ux} \]
14. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt
   (*
    (fma (- (* (fma -1.0 maxCos 2.0) ux) 2.0) maxCos (* (- (/ 2.0 ux) 1.0) ux))
    ux))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(fma(Float32(Float32(fma(Float32(-1.0), maxCos, Float32(2.0)) * ux) - Float32(2.0)), maxCos, Float32(Float32(Float32(Float32(2.0) / ux) - Float32(1.0)) * ux)) * ux)))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in ux around inf

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(\left(-1 \cdot {\left(maxCos - 1\right)}^{2} + 2 \cdot \frac{1}{ux}\right) - 2 \cdot \frac{maxCos}{ux}\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 98.2%

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

9. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(maxCos \cdot \left(-1 \cdot \left(maxCos \cdot ux\right) + ux \cdot \left(2 - 2 \cdot \frac{1}{ux}\right)\right) + ux \cdot \left(2 \cdot \frac{1}{ux} - 1\right)\right) \cdot ux} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. associate-*r* N/A

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

    4. mul-1-neg N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-neg.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. lower--.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 N/A

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

    13. *-commutative N/A

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in ux around 0

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

    1. lower--.f32 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f32 N/A

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

    4. mul-1-neg N/A

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

    5. +-commutative N/A

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

    6. mul-1-neg N/A

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

    7. lower-fma.f32 98.2

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

14. Applied rewrites 98.2%

    \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-1, maxCos, 2\right) \cdot ux - 2, maxCos, \left(\frac{2}{ux} - 1\right) \cdot ux\right) \cdot ux} \]
15. Add Preprocessing

Alternative 3: 98.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt
   (fma
    (fma (- (* ux 2.0) 2.0) ux (* (- maxCos) (* ux ux)))
    maxCos
    (* (fma -1.0 ux 2.0) ux)))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(fma(Float32(Float32(ux * Float32(2.0)) - Float32(2.0)), ux, Float32(Float32(-maxCos) * Float32(ux * ux))), maxCos, Float32(fma(Float32(-1.0), ux, Float32(2.0)) * ux))))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in ux around inf

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(\left(-1 \cdot {\left(maxCos - 1\right)}^{2} + 2 \cdot \frac{1}{ux}\right) - 2 \cdot \frac{maxCos}{ux}\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

8. Applied rewrites 98.2%

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

9. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(maxCos \cdot \left(-1 \cdot \left(maxCos \cdot ux\right) + ux \cdot \left(2 - 2 \cdot \frac{1}{ux}\right)\right) + ux \cdot \left(2 \cdot \frac{1}{ux} - 1\right)\right) \cdot ux} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. associate-*r* N/A

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

    4. mul-1-neg N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-neg.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. lower--.f32 N/A

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

    10. associate-*r/ N/A

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

    11. metadata-eval N/A

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

    12. lower-/.f32 N/A

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

    13. *-commutative N/A

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in maxCos around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. lower--.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. associate-*r* N/A

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

    10. mul-1-neg N/A

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

    11. lower-*.f32 N/A

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

    12. lift-neg.f32 N/A

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

    13. pow2 N/A

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

    14. lift-*.f32 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f32 N/A

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

14. Applied rewrites 98.3%

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

Alternative 4: 98.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt
   (fma
    (fma (- maxCos) (* ux ux) (* (- (* ux 2.0) 2.0) ux))
    maxCos
    (* (fma -1.0 ux 2.0) ux)))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(fma(Float32(-maxCos), Float32(ux * ux), Float32(Float32(Float32(ux * Float32(2.0)) - Float32(2.0)) * ux)), maxCos, Float32(fma(Float32(-1.0), ux, Float32(2.0)) * ux))))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

8. Applied rewrites 98.3%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt
   (*
    (- (fma (- ux) (fma (- maxCos 2.0) maxCos 1.0) 2.0) (* maxCos 2.0))
    ux))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(fma(Float32(-ux), fma(Float32(maxCos - Float32(2.0)), maxCos, Float32(1.0)), Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

8. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \mathsf{fma}\left(maxCos - 2, maxCos, 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
9. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt (* (+ (fma (- (* ux 2.0) 2.0) maxCos (- ux)) 2.0) ux))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(fma(Float32(Float32(ux * Float32(2.0)) - Float32(2.0)), maxCos, Float32(-ux)) + Float32(2.0)) * ux)))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 + \left(-1 \cdot ux + maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f32 97.6

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

8. Applied rewrites 97.6%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(ux \cdot 2 - 2, maxCos, -ux\right) + 2\right) \cdot ux} \]
9. Add Preprocessing

Alternative 7: 96.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) PI))
  (sqrt (* (- (fma -1.0 ux 2.0) (* maxCos 2.0)) ux))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(fma(Float32(-1.0), ux, Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 + -1 \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 96.9

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

8. Applied rewrites 96.9%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(-1, ux, 2\right) - maxCos \cdot 2\right) \cdot ux} \]
9. Add Preprocessing

Alternative 8: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999494757503e-5)
   (* (sin (* (+ uy uy) PI)) (sqrt (* (+ (- ux) 2.0) ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999494757503e-5f) {
		tmp = sinf(((uy + uy) * ((float) M_PI))) * sqrtf(((-ux + 2.0f) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999494757503e-5))
		tmp = Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(Float32(Float32(-ux) + Float32(2.0)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   1. Initial program 58.1%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 + -1 \cdot ux\right) \cdot ux} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f32 97.7

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

   8. Applied rewrites 97.7%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 97.7

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

   10. Applied rewrites 97.7%

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

       1. lift-fma.f32 N/A

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

       2. lower-+.f32 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(ux\right)\right) + 2\right) \cdot ux} \]

       4. lift-neg.f32 97.7

          \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \]

   12. Applied rewrites 97.7%

       \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \]

   if 1.99999995e-5 < maxCos

   1. Initial program 57.9%

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

   3. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - \left(\mathsf{neg}\left(-2\right)\right) \cdot maxCos\right)} \]

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

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.3

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

   5. Applied rewrites 76.3%

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

Alternative 9: 84.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;ux \leq 0.00019538949709385633:\\ \;\;\;\;\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= ux 0.00019538949709385633)
     (* (sin (* (+ uy uy) PI)) (sqrt (* 2.0 ux)))
     (*
      (*
       (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (* PI 2.0))
       uy)
      (sqrt (- 1.0 (* t_0 t_0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (ux <= 0.00019538949709385633f) {
		tmp = sinf(((uy + uy) * ((float) M_PI))) * sqrtf((2.0f * ux));
	} else {
		tmp = (fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * uy) * sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019538949709385633))
		tmp = Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * uy) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.95389497e-4

   1. Initial program 38.2%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 + -1 \cdot ux\right) \cdot ux} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f32 92.3

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

   8. Applied rewrites 92.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 92.3

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

   10. Applied rewrites 92.3%

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

   11. Taylor expanded in ux around 0

       \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux} \]
   12. Step-by-step derivation

   13. Applied rewrites 86.7%

       \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux} \]

   if 1.95389497e-4 < ux

   1. Initial program 89.6%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 82.1

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

   5. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), {\pi}^{3}, \pi \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. unpow3 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 82.1

         \[\leadsto \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. Applied rewrites 82.1%

      \[\leadsto \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 92.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sin (* (+ uy uy) PI)) (sqrt (* (+ (- ux) 2.0) ux))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy + uy) * ((float) M_PI))) * sqrtf(((-ux + 2.0f) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(Float32(Float32(-ux) + Float32(2.0)) * ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin(((uy + uy) * single(pi))) * sqrt(((-ux + single(2.0)) * ux));
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot {\left(maxCos - 1\right)}^{2} + 2\right) - 2 \cdot maxCos\right) \cdot ux} \]

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 + -1 \cdot ux\right) \cdot ux} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f32 92.4

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

8. Applied rewrites 92.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 92.4

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

10. Applied rewrites 92.4%

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

    1. lift-fma.f32 N/A

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

    2. lower-+.f32 N/A

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

    3. mul-1-neg N/A

       \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(ux\right)\right) + 2\right) \cdot ux} \]

    4. lift-neg.f32 92.4

       \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \]

12. Applied rewrites 92.4%

    \[\leadsto \sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \]
13. Add Preprocessing

Alternative 11: 54.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (*
    (* (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (* PI 2.0)) uy)
    (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return (fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * uy) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * uy) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 54.1

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

5. Applied rewrites 54.1%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), {\pi}^{3}, \pi \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. unpow3 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 54.1

      \[\leadsto \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

7. Applied rewrites 54.1%

   \[\leadsto \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
8. Add Preprocessing

Alternative 12: 51.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* PI (* 2.0 uy))
  (sqrt
   (-
    1.0
    (* (+ (- 1.0 ux) (* ux maxCos)) (* (- (+ (/ 1.0 ux) maxCos) 1.0) ux))))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * Float32(Float32(Float32(Float32(Float32(1.0) / ux) + maxCos) - Float32(1.0)) * ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(pi) * (single(2.0) * uy)) * sqrt((single(1.0) - (((single(1.0) - ux) + (ux * maxCos)) * ((((single(1.0) / ux) + maxCos) - single(1.0)) * ux))));
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(maxCos + \frac{1}{ux}\right) - 1\right) \cdot ux\right)} \]

   4. +-commutative N/A

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)} \]

   5. lower-+.f32 N/A

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)} \]

   6. lower-/.f32 51.8

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)} \]

8. Applied rewrites 51.8%

   \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}} \]
9. Add Preprocessing

Alternative 13: 51.2% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* PI (* 2.0 uy))
  (sqrt
   (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (+ (- 1.0 ux) (* ux maxCos)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 51.2

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

7. Applied rewrites 51.2%

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

Alternative 14: 51.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lift--.f32 51.2

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

8. Applied rewrites 51.2%

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

Alternative 15: 51.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux))))
   (* (* PI (* 2.0 uy)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	return (((float) M_PI) * (2.0f * uy)) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	return Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f32 N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-fma.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift-fma.f32 N/A

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

   14. lift--.f32 51.1

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

7. Applied rewrites 51.1%

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

Alternative 16: 49.8% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* PI (* 2.0 uy))
  (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (- 1.0 ux))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in maxCos around 0

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

   1. lift--.f32 49.8

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

8. Applied rewrites 49.8%

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

Alternative 17: 42.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 42.0

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

8. Applied rewrites 42.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 42.0

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

10. Applied rewrites 42.0%

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

Alternative 18: 41.4% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* PI (* 2.0 uy)) (sqrt (- 1.0 (fma -2.0 ux 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 42.0

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

8. Applied rewrites 42.0%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 41.4%

    \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(-2, ux, 1\right)} \]
12. Add Preprocessing

Alternative 19: 7.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* PI (* 2.0 uy)) (sqrt (- 1.0 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return (((float) M_PI) * (2.0f * uy)) * sqrtf((1.0f - 1.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) * sqrt(Float32(Float32(1.0) - Float32(1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(pi) * (single(2.0) * uy)) * sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

   \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 7.1%

   \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{1}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))