UniformSampleCone, x

Percentage Accurate: 57.3% → 99.0%

Time: 15.2s

Alternatives: 5

Speedup: N/A×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

Alternative 2: 98.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift-/.f32 88.5

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

8. Applied rewrites 88.5%

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

9. Taylor expanded in ux around inf

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

    1. lower-*.f32 N/A

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

    2. pow2 N/A

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

    3. lift-*.f32 N/A

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

    4. lower--.f32 N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-pow.f32 N/A

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

    7. lower-+.f32 N/A

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

    8. lower-*.f32 N/A

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

    9. lift-/.f32 N/A

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

    10. lift-*.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. lower-/.f32 99.0

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

11. Applied rewrites 99.0%

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

12. Final simplification 99.0%

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

Alternative 3: 96.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - maxCos \cdot 2\\ t_1 := {t\_0}^{2.5}\\ t_2 := {t\_0}^{1.5}\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{t\_2 \cdot t\_2}}, {\left(maxCos - 1\right)}^{4}, \left(-0.0625 \cdot \sqrt{\frac{\frac{1}{ux}}{t\_1 \cdot t\_1}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(-0.5 \cdot \sqrt{\frac{\frac{1}{ux}}{t\_0}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* maxCos 2.0))) (t_1 (pow t_0 2.5)) (t_2 (pow t_0 1.5)))
   (*
    (cos (* (* uy 2.0) PI))
    (fma
     (fma
      (fma
       (* -0.125 (sqrt (/ (/ 1.0 (pow ux 3.0)) (* t_2 t_2))))
       (pow (- maxCos 1.0) 4.0)
       (*
        (* -0.0625 (sqrt (/ (/ 1.0 ux) (* t_1 t_1))))
        (pow (- maxCos 1.0) 6.0)))
      (* ux ux)
      (* (* -0.5 (sqrt (/ (/ 1.0 ux) t_0))) (pow (- maxCos 1.0) 2.0)))
     (* ux ux)
     (sqrt (* ux t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f - (maxCos * 2.0f);
	float t_1 = powf(t_0, 2.5f);
	float t_2 = powf(t_0, 1.5f);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * fmaf(fmaf(fmaf((-0.125f * sqrtf(((1.0f / powf(ux, 3.0f)) / (t_2 * t_2)))), powf((maxCos - 1.0f), 4.0f), ((-0.0625f * sqrtf(((1.0f / ux) / (t_1 * t_1)))) * powf((maxCos - 1.0f), 6.0f))), (ux * ux), ((-0.5f * sqrtf(((1.0f / ux) / t_0))) * powf((maxCos - 1.0f), 2.0f))), (ux * ux), sqrtf((ux * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) - Float32(maxCos * Float32(2.0)))
	t_1 = t_0 ^ Float32(2.5)
	t_2 = t_0 ^ Float32(1.5)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * fma(fma(fma(Float32(Float32(-0.125) * sqrt(Float32(Float32(Float32(1.0) / (ux ^ Float32(3.0))) / Float32(t_2 * t_2)))), (Float32(maxCos - Float32(1.0)) ^ Float32(4.0)), Float32(Float32(Float32(-0.0625) * sqrt(Float32(Float32(Float32(1.0) / ux) / Float32(t_1 * t_1)))) * (Float32(maxCos - Float32(1.0)) ^ Float32(6.0)))), Float32(ux * ux), Float32(Float32(Float32(-0.5) * sqrt(Float32(Float32(Float32(1.0) / ux) / t_0))) * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0)))), Float32(ux * ux), sqrt(Float32(ux * t_0))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + {ux}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \cdot {\left(maxCos - 1\right)}^{2}\right) + {ux}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\sqrt{\frac{1}{{ux}^{3} \cdot {\left(2 - 2 \cdot maxCos\right)}^{3}}} \cdot {\left(maxCos - 1\right)}^{4}\right) + \frac{-1}{16} \cdot \left(\sqrt{\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}} \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)} \]

5. Applied rewrites 97.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{1.5} \cdot {\left(2 - maxCos \cdot 2\right)}^{1.5}}}, {\left(maxCos - 1\right)}^{4}, \left(-0.0625 \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{2.5} \cdot {\left(2 - maxCos \cdot 2\right)}^{2.5}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(-0.5 \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux} \cdot \sqrt{2 - maxCos \cdot 2}\right)} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8} \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}}}}, {\left(maxCos - 1\right)}^{4}, \left(\frac{-1}{16} \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(\frac{-1}{2} \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux} \cdot \sqrt{2 - maxCos \cdot 2}\right) \]

   2. lift-sqrt.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8} \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}}}}, {\left(maxCos - 1\right)}^{4}, \left(\frac{-1}{16} \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(\frac{-1}{2} \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux} \cdot \sqrt{2 - maxCos \cdot 2}\right) \]

   3. lift-sqrt.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8} \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}}}}, {\left(maxCos - 1\right)}^{4}, \left(\frac{-1}{16} \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(\frac{-1}{2} \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux} \cdot \sqrt{2 - maxCos \cdot 2}\right) \]

   4. sqrt-unprod N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8} \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}}}}, {\left(maxCos - 1\right)}^{4}, \left(\frac{-1}{16} \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(\frac{-1}{2} \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\right) \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8} \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{3}{2}}}}, {\left(maxCos - 1\right)}^{4}, \left(\frac{-1}{16} \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}} \cdot {\left(2 - maxCos \cdot 2\right)}^{\frac{5}{2}}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(\frac{-1}{2} \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\right) \]

   6. lower-*.f32 97.4

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{1.5} \cdot {\left(2 - maxCos \cdot 2\right)}^{1.5}}}, {\left(maxCos - 1\right)}^{4}, \left(-0.0625 \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{2.5} \cdot {\left(2 - maxCos \cdot 2\right)}^{2.5}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(-0.5 \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\right) \]

7. Applied rewrites 97.4%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125 \cdot \sqrt{\frac{\frac{1}{{ux}^{3}}}{{\left(2 - maxCos \cdot 2\right)}^{1.5} \cdot {\left(2 - maxCos \cdot 2\right)}^{1.5}}}, {\left(maxCos - 1\right)}^{4}, \left(-0.0625 \cdot \sqrt{\frac{\frac{1}{ux}}{{\left(2 - maxCos \cdot 2\right)}^{2.5} \cdot {\left(2 - maxCos \cdot 2\right)}^{2.5}}}\right) \cdot {\left(maxCos - 1\right)}^{6}\right), ux \cdot ux, \left(-0.5 \cdot \sqrt{\frac{\frac{1}{ux}}{2 - maxCos \cdot 2}}\right) \cdot {\left(maxCos - 1\right)}^{2}\right), ux \cdot ux, \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\right) \]
8. Add Preprocessing

Alternative 4: 96.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - 2 \cdot maxCos\\ t_1 := ux \cdot t\_0\\ t_2 := {t\_1}^{0.5}\\ t_3 := \sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right)\\ \mathsf{fma}\left(t\_2, \sin \left(uy \cdot \mathsf{fma}\left(0.5, \frac{\pi}{uy}, 2 \cdot \pi\right)\right), \left(ux \cdot ux\right) \cdot \left(-0.5 \cdot \left(\frac{1}{t\_2} \cdot \left(t\_3 \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.125, {\left(\frac{1}{{t\_1}^{3}}\right)}^{0.5} \cdot \left(t\_3 \cdot {\left(maxCos - 1\right)}^{4}\right), -0.0625 \cdot \left({\left(\frac{1}{ux \cdot {t\_0}^{5}}\right)}^{0.5} \cdot \left(t\_3 \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* 2.0 maxCos)))
        (t_1 (* ux t_0))
        (t_2 (pow t_1 0.5))
        (t_3 (sin (fma 2.0 (* uy PI) (/ PI 2.0)))))
   (fma
    t_2
    (sin (* uy (fma 0.5 (/ PI uy) (* 2.0 PI))))
    (*
     (* ux ux)
     (+
      (* -0.5 (* (/ 1.0 t_2) (* t_3 (pow (- maxCos 1.0) 2.0))))
      (*
       (* ux ux)
       (fma
        -0.125
        (* (pow (/ 1.0 (pow t_1 3.0)) 0.5) (* t_3 (pow (- maxCos 1.0) 4.0)))
        (*
         -0.0625
         (*
          (pow (/ 1.0 (* ux (pow t_0 5.0))) 0.5)
          (* t_3 (pow (- maxCos 1.0) 6.0)))))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f - (2.0f * maxCos);
	float t_1 = ux * t_0;
	float t_2 = powf(t_1, 0.5f);
	float t_3 = sinf(fmaf(2.0f, (uy * ((float) M_PI)), (((float) M_PI) / 2.0f)));
	return fmaf(t_2, sinf((uy * fmaf(0.5f, (((float) M_PI) / uy), (2.0f * ((float) M_PI))))), ((ux * ux) * ((-0.5f * ((1.0f / t_2) * (t_3 * powf((maxCos - 1.0f), 2.0f)))) + ((ux * ux) * fmaf(-0.125f, (powf((1.0f / powf(t_1, 3.0f)), 0.5f) * (t_3 * powf((maxCos - 1.0f), 4.0f))), (-0.0625f * (powf((1.0f / (ux * powf(t_0, 5.0f))), 0.5f) * (t_3 * powf((maxCos - 1.0f), 6.0f)))))))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))
	t_1 = Float32(ux * t_0)
	t_2 = t_1 ^ Float32(0.5)
	t_3 = sin(fma(Float32(2.0), Float32(uy * Float32(pi)), Float32(Float32(pi) / Float32(2.0))))
	return fma(t_2, sin(Float32(uy * fma(Float32(0.5), Float32(Float32(pi) / uy), Float32(Float32(2.0) * Float32(pi))))), Float32(Float32(ux * ux) * Float32(Float32(Float32(-0.5) * Float32(Float32(Float32(1.0) / t_2) * Float32(t_3 * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) + Float32(Float32(ux * ux) * fma(Float32(-0.125), Float32((Float32(Float32(1.0) / (t_1 ^ Float32(3.0))) ^ Float32(0.5)) * Float32(t_3 * (Float32(maxCos - Float32(1.0)) ^ Float32(4.0)))), Float32(Float32(-0.0625) * Float32((Float32(Float32(1.0) / Float32(ux * (t_0 ^ Float32(5.0)))) ^ Float32(0.5)) * Float32(t_3 * (Float32(maxCos - Float32(1.0)) ^ Float32(6.0))))))))))
end

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + {ux}^{2} \cdot \left(\frac{-1}{2} \cdot \left(\sqrt{\frac{1}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + {ux}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\sqrt{\frac{1}{{ux}^{3} \cdot {\left(2 - 2 \cdot maxCos\right)}^{3}}} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right)\right) + \frac{-1}{16} \cdot \left(\sqrt{\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)} \]

8. Applied rewrites 97.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{0.5}, \sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right), \left(ux \cdot ux\right) \cdot \left(-0.5 \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{0.5}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.125, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{0.5} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), -0.0625 \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{0.5} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right)} \]

9. Taylor expanded in uy around inf

   \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}, \sin \left(uy \cdot \left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{uy} + 2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(ux \cdot ux\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), \frac{-1}{16} \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}, \sin \left(uy \cdot \left(\frac{1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{uy} + 2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(ux \cdot ux\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), \frac{-1}{16} \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]

    2. lower-fma.f32 N/A

       \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}, \sin \left(uy \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{uy}, 2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(ux \cdot ux\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), \frac{-1}{16} \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]

    3. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}, \sin \left(uy \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{PI}\left(\right)}{uy}, 2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(ux \cdot ux\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), \frac{-1}{16} \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]

    4. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}, \sin \left(uy \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\pi}{uy}, 2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(ux \cdot ux\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), \frac{-1}{16} \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]

    5. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}, \sin \left(uy \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\pi}{uy}, 2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(ux \cdot ux\right) \cdot \left(\frac{-1}{2} \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{\frac{1}{2}}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(\frac{-1}{8}, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), \frac{-1}{16} \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{\frac{1}{2}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]

    6. lift-PI.f32 97.2

       \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{0.5}, \sin \left(uy \cdot \mathsf{fma}\left(0.5, \frac{\pi}{uy}, 2 \cdot \pi\right)\right), \left(ux \cdot ux\right) \cdot \left(-0.5 \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{0.5}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.125, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{0.5} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), -0.0625 \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{0.5} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]

11. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left({\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{0.5}, \sin \left(uy \cdot \mathsf{fma}\left(0.5, \frac{\pi}{uy}, 2 \cdot \pi\right)\right), \left(ux \cdot ux\right) \cdot \left(-0.5 \cdot \left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{0.5}} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(ux \cdot ux\right) \cdot \mathsf{fma}\left(-0.125, {\left(\frac{1}{{\left(ux \cdot \left(2 - 2 \cdot maxCos\right)\right)}^{3}}\right)}^{0.5} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{4}\right), -0.0625 \cdot \left({\left(\frac{1}{ux \cdot {\left(2 - 2 \cdot maxCos\right)}^{5}}\right)}^{0.5} \cdot \left(\sin \left(\mathsf{fma}\left(2, uy \cdot \pi, \frac{\pi}{2}\right)\right) \cdot {\left(maxCos - 1\right)}^{6}\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 76.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 78.4%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))