UniformSampleCone, x

Percentage Accurate: 57.7% → 98.9%

Time: 43.5s

Alternatives: 8

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: 57.7% 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: 98.9% accurate, N/A× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt (* (- (fma (- ux) (pow (- maxCos 1.0) 2.0) 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(-ux, powf((maxCos - 1.0f), 2.0f), 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(-ux), (Float32(maxCos - Float32(1.0)) ^ Float32(2.0)), Float32(2.0)) - Float32(maxCos * Float32(2.0))) * ux)))
end

Derivation

1. Initial program 57.7%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

Alternative 2: 98.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower--.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 98.9

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

5. Applied rewrites 98.9%

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

Alternative 3: 95.9% accurate, N/A× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* 2.0 maxCos))) (t_1 (* ux t_0)))
   (*
    (cos (* (* uy 2.0) PI))
    (+
     (sqrt t_1)
     (*
      (* ux ux)
      (fma
       -0.5
       (* (sqrt (/ 1.0 t_1)) (pow (- maxCos 1.0) 2.0))
       (*
        (* ux ux)
        (fma
         -0.125
         (* (sqrt (pow t_1 -3.0)) (pow (- maxCos 1.0) 4.0))
         (*
          -0.0625
          (*
           (sqrt (/ 1.0 (* ux (pow t_0 5.0))))
           (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;
	return cosf(((uy * 2.0f) * ((float) M_PI))) * (sqrtf(t_1) + ((ux * ux) * fmaf(-0.5f, (sqrtf((1.0f / t_1)) * powf((maxCos - 1.0f), 2.0f)), ((ux * ux) * fmaf(-0.125f, (sqrtf(powf(t_1, -3.0f)) * powf((maxCos - 1.0f), 4.0f)), (-0.0625f * (sqrtf((1.0f / (ux * powf(t_0, 5.0f)))) * 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)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * Float32(sqrt(t_1) + Float32(Float32(ux * ux) * fma(Float32(-0.5), Float32(sqrt(Float32(Float32(1.0) / t_1)) * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))), Float32(Float32(ux * ux) * fma(Float32(-0.125), Float32(sqrt((t_1 ^ Float32(-3.0))) * (Float32(maxCos - Float32(1.0)) ^ Float32(4.0))), Float32(Float32(-0.0625) * Float32(sqrt(Float32(Float32(1.0) / Float32(ux * (t_0 ^ Float32(5.0))))) * (Float32(maxCos - Float32(1.0)) ^ Float32(6.0))))))))))
end

Derivation

1. Initial program 57.7%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 95.7%

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

10. Applied rewrites 95.9%

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

Alternative 4: 95.9% accurate, N/A× speedup

Math

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

FPCore

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

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

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

Derivation

1. Initial program 57.7%

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

   1. lift-cos.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. cos-2 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-cos.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-cos.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lower-*.f32 N/A

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

4. Applied rewrites 57.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 95.9%

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

Alternative 5: 90.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot t\_0\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* 2.0 uy)))))
   (fma
    (*
     (* (pow (- maxCos 1.0) 2.0) t_0)
     (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
    -0.5
    (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (2.0f * uy)));
	return fmaf(((powf((maxCos - 1.0f), 2.0f) * t_0) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))), -0.5f, (t_0 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	return fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * t_0) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32(-0.5), Float32(t_0 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))))
end

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]
6. Add Preprocessing

Alternative 6: 89.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.08888888888888889, \left(uy \cdot uy\right) \cdot {\pi}^{6}, 0.6666666666666666 \cdot {\pi}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (fma
  (*
   (*
    (pow (- maxCos 1.0) 2.0)
    (+
     1.0
     (*
      (* uy uy)
      (fma
       -2.0
       (* PI PI)
       (*
        (* uy uy)
        (fma
         -0.08888888888888889
         (* (* uy uy) (pow PI 6.0))
         (* 0.6666666666666666 (pow PI 4.0))))))))
   (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
  -0.5
  (*
   (sin (fma PI (* 2.0 uy) (/ PI 2.0)))
   (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	return fmaf(((powf((maxCos - 1.0f), 2.0f) * (1.0f + ((uy * uy) * fmaf(-2.0f, (((float) M_PI) * ((float) M_PI)), ((uy * uy) * fmaf(-0.08888888888888889f, ((uy * uy) * powf(((float) M_PI), 6.0f)), (0.6666666666666666f * powf(((float) M_PI), 4.0f)))))))) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))), -0.5f, (sinf(fmaf(((float) M_PI), (2.0f * uy), (((float) M_PI) / 2.0f))) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux))));
}

Julia

function code(ux, uy, maxCos)
	return fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * Float32(Float32(1.0) + Float32(Float32(uy * uy) * fma(Float32(-2.0), Float32(Float32(pi) * Float32(pi)), Float32(Float32(uy * uy) * fma(Float32(-0.08888888888888889), Float32(Float32(uy * uy) * (Float32(pi) ^ Float32(6.0))), Float32(Float32(0.6666666666666666) * (Float32(pi) ^ Float32(4.0))))))))) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32(-0.5), Float32(sin(fma(Float32(pi), Float32(Float32(2.0) * uy), Float32(Float32(pi) / Float32(2.0)))) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))))
end

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.4%

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

   1. lift-cos.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

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

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

   6. lower-sin.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 90.3

       \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

7. Applied rewrites 90.3%

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + {uy}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + {uy}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + {uy}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + {uy}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   4. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   6. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, {\mathsf{PI}\left(\right)}^{2}, {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   8. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   9. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \pi \cdot \mathsf{PI}\left(\right), {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   10. lift-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{3} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]

10. Applied rewrites 89.7%

    \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \left(1 + \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \pi \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.08888888888888889, \left(uy \cdot uy\right) \cdot {\pi}^{6}, 0.6666666666666666 \cdot {\pi}^{4}\right)\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sin \left(\mathsf{fma}\left(\pi, 2 \cdot uy, \frac{\pi}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]
11. Add Preprocessing

Alternative 7: 88.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot 1\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (fma
  (*
   (* (pow (- maxCos 1.0) 2.0) 1.0)
   (sqrt (/ (pow ux 3.0) (fma -2.0 maxCos 2.0))))
  -0.5
  (* (cos (* PI (* 2.0 uy))) (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	return fmaf(((powf((maxCos - 1.0f), 2.0f) * 1.0f) * sqrtf((powf(ux, 3.0f) / fmaf(-2.0f, maxCos, 2.0f)))), -0.5f, (cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux))));
}

Julia

function code(ux, uy, maxCos)
	return fma(Float32(Float32((Float32(maxCos - Float32(1.0)) ^ Float32(2.0)) * Float32(1.0)) * sqrt(Float32((ux ^ Float32(3.0)) / fma(Float32(-2.0), maxCos, Float32(2.0))))), Float32(-0.5), Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux))))
end

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

6. Taylor expanded in uy around 0

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot 1\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, \frac{-1}{2}, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]
7. Step-by-step derivation

8. Applied rewrites 88.1%

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

9. Final simplification 88.1%

   \[\leadsto \mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot 1\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right) \]
10. Add Preprocessing

Alternative 8: 85.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(maxCos - 1\right)}^{2}\\ t_1 := 2 + -2 \cdot maxCos\\ t_2 := \sqrt{ux \cdot t\_1}\\ t_3 := \sqrt{\frac{{ux}^{3}}{t\_1}}\\ t\_2 + \mathsf{fma}\left(-0.5, t\_3 \cdot t\_0, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, t\_2 \cdot \left(\pi \cdot \pi\right), \mathsf{fma}\left(t\_3, {\left(\pi \cdot \left(maxCos - 1\right)\right)}^{2}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.3333333333333333, t\_3 \cdot \left({\pi}^{4} \cdot t\_0\right), \mathsf{fma}\left(0.6666666666666666, t\_2 \cdot {\pi}^{4}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.08888888888888889, t\_2 \cdot {\pi}^{6}, 0.044444444444444446 \cdot \left(t\_3 \cdot \left({\pi}^{6} \cdot t\_0\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (pow (- maxCos 1.0) 2.0))
        (t_1 (+ 2.0 (* -2.0 maxCos)))
        (t_2 (sqrt (* ux t_1)))
        (t_3 (sqrt (/ (pow ux 3.0) t_1))))
   (+
    t_2
    (fma
     -0.5
     (* t_3 t_0)
     (*
      (* uy uy)
      (fma
       -2.0
       (* t_2 (* PI PI))
       (fma
        t_3
        (pow (* PI (- maxCos 1.0)) 2.0)
        (*
         (* uy uy)
         (fma
          -0.3333333333333333
          (* t_3 (* (pow PI 4.0) t_0))
          (fma
           0.6666666666666666
           (* t_2 (pow PI 4.0))
           (*
            (* uy uy)
            (fma
             -0.08888888888888889
             (* t_2 (pow PI 6.0))
             (* 0.044444444444444446 (* t_3 (* (pow PI 6.0) t_0)))))))))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = powf((maxCos - 1.0f), 2.0f);
	float t_1 = 2.0f + (-2.0f * maxCos);
	float t_2 = sqrtf((ux * t_1));
	float t_3 = sqrtf((powf(ux, 3.0f) / t_1));
	return t_2 + fmaf(-0.5f, (t_3 * t_0), ((uy * uy) * fmaf(-2.0f, (t_2 * (((float) M_PI) * ((float) M_PI))), fmaf(t_3, powf((((float) M_PI) * (maxCos - 1.0f)), 2.0f), ((uy * uy) * fmaf(-0.3333333333333333f, (t_3 * (powf(((float) M_PI), 4.0f) * t_0)), fmaf(0.6666666666666666f, (t_2 * powf(((float) M_PI), 4.0f)), ((uy * uy) * fmaf(-0.08888888888888889f, (t_2 * powf(((float) M_PI), 6.0f)), (0.044444444444444446f * (t_3 * (powf(((float) M_PI), 6.0f) * t_0))))))))))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(maxCos - Float32(1.0)) ^ Float32(2.0)
	t_1 = Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos))
	t_2 = sqrt(Float32(ux * t_1))
	t_3 = sqrt(Float32((ux ^ Float32(3.0)) / t_1))
	return Float32(t_2 + fma(Float32(-0.5), Float32(t_3 * t_0), Float32(Float32(uy * uy) * fma(Float32(-2.0), Float32(t_2 * Float32(Float32(pi) * Float32(pi))), fma(t_3, (Float32(Float32(pi) * Float32(maxCos - Float32(1.0))) ^ Float32(2.0)), Float32(Float32(uy * uy) * fma(Float32(-0.3333333333333333), Float32(t_3 * Float32((Float32(pi) ^ Float32(4.0)) * t_0)), fma(Float32(0.6666666666666666), Float32(t_2 * (Float32(pi) ^ Float32(4.0))), Float32(Float32(uy * uy) * fma(Float32(-0.08888888888888889), Float32(t_2 * (Float32(pi) ^ Float32(6.0))), Float32(Float32(0.044444444444444446) * Float32(t_3 * Float32((Float32(pi) ^ Float32(6.0)) * t_0)))))))))))))
end

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

5. Applied rewrites 90.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(maxCos - 1\right)}^{2} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

6. Taylor expanded in uy around 0

   \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} + \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot {\left(maxCos - 1\right)}^{2}\right) + {uy}^{2} \cdot \left(-2 \cdot \left(\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(maxCos - 1\right)}^{2}\right) + {uy}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\mathsf{PI}\left(\right)}^{4} \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\frac{2}{3} \cdot \left(\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + {uy}^{2} \cdot \left(\frac{-4}{45} \cdot \left(\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\mathsf{PI}\left(\right)}^{6}\right) + \frac{2}{45} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\mathsf{PI}\left(\right)}^{6} \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)} \]

8. Applied rewrites 85.6%

   \[\leadsto \mathsf{fma}\left(\sqrt{ux}, \color{blue}{\sqrt{2 + -2 \cdot maxCos}}, \mathsf{fma}\left(-0.5, \sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot {\left(maxCos - 1\right)}^{2}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot \left(\pi \cdot \pi\right), \mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}}, {\left(\pi \cdot \left(maxCos - 1\right)\right)}^{2}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\pi}^{4} \cdot {\left(maxCos - 1\right)}^{2}\right), \mathsf{fma}\left(0.6666666666666666, \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\pi}^{4}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.08888888888888889, \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\pi}^{6}, 0.044444444444444446 \cdot \left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\pi}^{6} \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Applied rewrites 85.6%

    \[\leadsto \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} + \mathsf{fma}\left(-0.5, \color{blue}{\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot {\left(maxCos - 1\right)}^{2}}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-2, \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot \left(\pi \cdot \pi\right), \mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}}, {\left(\pi \cdot \left(maxCos - 1\right)\right)}^{2}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.3333333333333333, \sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\pi}^{4} \cdot {\left(maxCos - 1\right)}^{2}\right), \mathsf{fma}\left(0.6666666666666666, \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\pi}^{4}, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.08888888888888889, \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \cdot {\pi}^{6}, 0.044444444444444446 \cdot \left(\sqrt{\frac{{ux}^{3}}{2 + -2 \cdot maxCos}} \cdot \left({\pi}^{6} \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))))