UniformSampleCone, x

Percentage Accurate: 57.3% → 99.1%

Time: 7.1s

Alternatives: 15

Speedup: 12.1×

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

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

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

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

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: 99.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sin (fma (- uy) (+ PI PI) (* 0.5 PI)))
 (sqrt
  (*
   ux
   (-
    (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0))))
    (* 2.0 maxCos))))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(-uy), Float32(Float32(pi) + Float32(pi)), Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos)))))
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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. count-2 N/A

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

   7. lift-+.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

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

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

   11. lower-sin.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. distribute-lft-neg-in N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-neg.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. mult-flip-rev N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 99.1%

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

6. Applied rewrites 99.1%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sin (fma -2.0 (* uy PI) (* 0.5 PI)))
 (sqrt
  (*
   ux
   (-
    (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0))))
    (* 2.0 maxCos))))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(-2.0), Float32(uy * Float32(pi)), Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos)))))
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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. count-2 N/A

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

   7. lift-+.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

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

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

   11. lower-sin.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. distribute-lft-neg-in N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-neg.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. mult-flip-rev N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f32 99.1%

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in uy around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 99.0%

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

9. Applied rewrites 99.0%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(fma(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos - Float32(1.0))), ux, Float32(2.0)) - maxCos) - maxCos))))
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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.8%

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

7. Applied rewrites 98.8%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. associate--r+ N/A

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

   5. lower--.f32 N/A

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

9. Applied rewrites 98.9%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(t\_0 - 0\right) \cdot \left(-\left(t\_0 - 2\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (- ux (* maxCos ux))))
  (* (cos (* (* uy 2.0) PI)) (sqrt (* (- t_0 0.0) (- (- t_0 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((t_0 - 0.0f) * -(t_0 - 2.0f)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(t_0 - Float32(0.0)) * Float32(-Float32(t_0 - Float32(2.0))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt(((t_0 - single(0.0)) * -(t_0 - single(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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-sqr-1 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 99.0%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]
4. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) + fma(Float32(-1.0), ux, Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - 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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 98.4%

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

7. Applied rewrites 98.4%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((ux * ((single(2.0) + (single(-1.0) * ux)) - (single(2.0) * maxCos))));
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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 97.6%

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

7. Applied rewrites 97.6%

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

Alternative 7: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{\left(2 - \mathsf{fma}\left(ux \cdot \left(maxCos - 1\right), maxCos - 1, maxCos + maxCos\right)\right) \cdot ux} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \pi, \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(\left(-uy\right) + \left(\left(-uy\right) - -0.5\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.0035000001080334187)
  (*
   (sqrt
    (*
     (-
      2.0
      (fma (* ux (- maxCos 1.0)) (- maxCos 1.0) (+ maxCos maxCos)))
     ux))
   (fma (* (* (* uy uy) -2.0) PI) PI 1.0))
  (*
   (sin (* PI (+ (- uy) (- (- uy) -0.5))))
   (sqrt (* ux (+ 2.0 (* -1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0035000001080334187f) {
		tmp = sqrtf(((2.0f - fmaf((ux * (maxCos - 1.0f)), (maxCos - 1.0f), (maxCos + maxCos))) * ux)) * fmaf((((uy * uy) * -2.0f) * ((float) M_PI)), ((float) M_PI), 1.0f);
	} else {
		tmp = sinf((((float) M_PI) * (-uy + (-uy - -0.5f)))) * sqrtf((ux * (2.0f + (-1.0f * ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - fma(Float32(ux * Float32(maxCos - Float32(1.0))), Float32(maxCos - Float32(1.0)), Float32(maxCos + maxCos))) * ux)) * fma(Float32(Float32(Float32(uy * uy) * Float32(-2.0)) * Float32(pi)), Float32(pi), Float32(1.0)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(-uy) + Float32(Float32(-uy) - Float32(-0.5))))) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(-1.0) * ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00350000011

   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. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 99.0%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-pow.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 88.5%

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

   7. Applied rewrites 88.5%

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

   9. Applied rewrites 88.5%

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

   if 0.00350000011 < uy

   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. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 99.0%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 92.9%

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

   7. Applied rewrites 92.9%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-neg.f32 N/A

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

      9. count-2-rev N/A

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

      10. distribute-lft-out N/A

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

      11. lift-PI.f32 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. associate-+r+ N/A

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

      17. lift-fma.f32 N/A

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

      18. lift-fma.f32 N/A

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

   9. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(\left(-uy\right) + \left(\left(-uy\right) - -0.5\right)\right)\right)} \cdot \sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{\left(2 - \mathsf{fma}\left(ux \cdot \left(maxCos - 1\right), maxCos - 1, maxCos + maxCos\right)\right) \cdot ux} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \pi, \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.0035000001080334187)
  (*
   (sqrt
    (*
     (-
      2.0
      (fma (* ux (- maxCos 1.0)) (- maxCos 1.0) (+ maxCos maxCos)))
     ux))
   (fma (* (* (* uy uy) -2.0) PI) PI 1.0))
  (* (sqrt (* (- 2.0 ux) ux)) (cos (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0035000001080334187f) {
		tmp = sqrtf(((2.0f - fmaf((ux * (maxCos - 1.0f)), (maxCos - 1.0f), (maxCos + maxCos))) * ux)) * fmaf((((uy * uy) * -2.0f) * ((float) M_PI)), ((float) M_PI), 1.0f);
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * cosf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0035000001080334187))
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - fma(Float32(ux * Float32(maxCos - Float32(1.0))), Float32(maxCos - Float32(1.0)), Float32(maxCos + maxCos))) * ux)) * fma(Float32(Float32(Float32(uy * uy) * Float32(-2.0)) * Float32(pi)), Float32(pi), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * cos(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00350000011

   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. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 99.0%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-pow.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 88.5%

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

   7. Applied rewrites 88.5%

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

   9. Applied rewrites 88.5%

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

   if 0.00350000011 < uy

   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. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 99.0%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 92.9%

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

   7. Applied rewrites 92.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 92.9%

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

   9. Applied rewrites 92.9%

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

Alternative 9: 88.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(Float32(2.0) - fma(Float32(ux * Float32(maxCos - Float32(1.0))), Float32(maxCos - Float32(1.0)), Float32(maxCos + maxCos))) * ux)) * fma(Float32(Float32(Float32(uy * uy) * Float32(-2.0)) * Float32(pi)), Float32(pi), Float32(1.0)))
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. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 88.5%

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

7. Applied rewrites 88.5%

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

9. Applied rewrites 88.5%

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

Alternative 10: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \sqrt{2 \cdot t\_0 - {t\_0}^{2}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (- ux (* maxCos ux))))
  (sqrt (- (* 2.0 t_0) (pow t_0 2.0)))))

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    t_0 = ux - (maxcos * ux)
    code = sqrt(((2.0e0 * t_0) - (t_0 ** 2.0e0)))
end function

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = sqrt(((single(2.0) * t_0) - (t_0 ^ single(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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. associate-+l- N/A

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

   7. sub-square-pow N/A

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

   8. associate--r+ N/A

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

   9. lower--.f32 N/A

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

3. Applied rewrites 59.0%

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

4. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}}} \]
5. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   2. lower--.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   3. lower-*.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   6. lower-pow.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   7. lower--.f32 N/A

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

   8. lower-*.f32 80.3%

      \[\leadsto \sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}} \]

6. Applied rewrites 80.3%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(ux - maxCos \cdot ux\right)}^{2}}} \]
7. Add Preprocessing

Alternative 11: 76.0% accurate, 2.6× speedup

Math

\[\sqrt{-\mathsf{fma}\left(-2, ux, {ux}^{2}\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (sqrt (- (fma -2.0 ux (pow ux 2.0)))))

C

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

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(-fma(Float32(-2.0), ux, (ux ^ 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. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.3%

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

4. Applied rewrites 49.3%

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

   1. lift-pow.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-square-pow N/A

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

   4. +-commutative N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f32 N/A

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

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

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

   8. unpow2 N/A

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

   9. lower-fma.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-fma.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lift-fma.f32 N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f32 N/A

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

   20. lower-*.f32 51.4%

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

6. Applied rewrites 51.4%

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

7. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\left(-2 \cdot ux + {ux}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(-2 \cdot ux + {ux}^{2}\right)\right)} \]

   2. lower-neg.f32 N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \sqrt{-\mathsf{fma}\left(-2, ux, {ux}^{2}\right)} \]

   4. lower-pow.f32 76.0%

      \[\leadsto \sqrt{-\mathsf{fma}\left(-2, ux, {ux}^{2}\right)} \]

9. Applied rewrites 76.0%

   \[\leadsto \sqrt{-\mathsf{fma}\left(-2, ux, {ux}^{2}\right)} \]
10. Add Preprocessing

Alternative 12: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.01850000023841858:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right), ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.01850000023841858)
    (sqrt (* (fma -2.0 maxCos 2.0) ux))
    (sqrt
     (fma
      (fma maxCos ux (- 1.0 ux))
      (- ux (fma maxCos ux 1.0))
      1.0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.01850000023841858f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	} else {
		tmp = sqrtf(fmaf(fmaf(maxCos, ux, (1.0f - ux)), (ux - fmaf(maxCos, ux, 1.0f)), 1.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.01850000023841858))
		tmp = sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux));
	else
		tmp = sqrt(fma(fma(maxCos, ux, Float32(Float32(1.0) - ux)), Float32(ux - fma(maxCos, ux, Float32(1.0))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0185000002

   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. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.3%

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 64.9%

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

   7. Applied rewrites 64.9%

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 64.9%

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

      4. lift--.f32 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 0.0185000002 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   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. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.3%

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

   4. Applied rewrites 49.3%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-square-pow N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f32 N/A

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

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

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

      8. unpow2 N/A

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

      9. lower-fma.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-fma.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-fma.f32 N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f32 N/A

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

      20. lower-*.f32 51.4%

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

   6. Applied rewrites 51.4%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

   8. Applied rewrites 49.5%

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

Alternative 13: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.01850000023841858:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.01850000023841858)
    (sqrt (* (fma -2.0 maxCos 2.0) ux))
    (sqrt
     (fma
      (- (fma maxCos ux 1.0) ux)
      (- ux (fma maxCos ux 1.0))
      1.0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.01850000023841858f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	} else {
		tmp = sqrtf(fmaf((fmaf(maxCos, ux, 1.0f) - ux), (ux - fmaf(maxCos, ux, 1.0f)), 1.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.01850000023841858))
		tmp = sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux));
	else
		tmp = sqrt(fma(Float32(fma(maxCos, ux, Float32(1.0)) - ux), Float32(ux - fma(maxCos, ux, Float32(1.0))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0185000002

   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. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.3%

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 64.9%

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

   7. Applied rewrites 64.9%

      \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
   8. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 64.9%

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

      4. lift--.f32 N/A

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

      5. sub-flip N/A

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

      6. +-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 0.0185000002 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   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. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.3%

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

   4. Applied rewrites 49.3%

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

      1. lift--.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. unpow2 N/A

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

      4. lift--.f32 N/A

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

      5. sub-negate-rev N/A

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

      6. lift-+.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-fma.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. +-commutative N/A

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

      14. associate-+r- N/A

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

      15. lift--.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. lift-fma.f32 N/A

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

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

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

      19. +-commutative N/A

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

   6. Applied rewrites 49.4%

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

Alternative 14: 64.9% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(fma(Float32(-2.0), 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. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.3%

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

4. Applied rewrites 49.3%

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

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 64.9%

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

7. Applied rewrites 64.9%

   \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 64.9%

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

   4. lift--.f32 N/A

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

   5. sub-flip N/A

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

   6. +-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 64.9%

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

9. Applied rewrites 64.9%

   \[\leadsto \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
10. Add Preprocessing

Alternative 15: 62.4% accurate, 12.1× speedup

Math

\[\sqrt{ux + ux} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (sqrt (+ ux ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux + ux));
}

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux + ux))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux + ux))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux + 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. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.3%

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

4. Applied rewrites 49.3%

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

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 64.9%

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

7. Applied rewrites 64.9%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{2 \cdot ux} \]
9. Step-by-step derivation

   1. lower-*.f32 62.4%

      \[\leadsto \sqrt{2 \cdot ux} \]

10. Applied rewrites 62.4%

    \[\leadsto \sqrt{2 \cdot ux} \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \sqrt{2 \cdot ux} \]

    2. count-2-rev N/A

       \[\leadsto \sqrt{ux + ux} \]

    3. lower-+.f32 62.4%

       \[\leadsto \sqrt{ux + ux} \]

12. Applied rewrites 62.4%

    \[\leadsto \sqrt{ux + ux} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025323 
(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)))))))