UniformSampleCone, x

Percentage Accurate: 57.6% → 99.0%

Time: 9.1s

Alternatives: 25

Speedup: 1.1×

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: 57.6% 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: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

Alternative 2: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

8. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

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

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

   6. distribute-rgt-in N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-fma.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. sub-negate-rev N/A

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

   15. lower--.f32 99.0%

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

10. Applied rewrites 99.0%

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

Alternative 3: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

8. Applied rewrites 99.0%

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

Alternative 4: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

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(Float32(Float32(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos))) * ux) * ux)))
end

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in uy around 0

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

    1. lower-fma.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 98.9%

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

11. Applied rewrites 98.9%

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

Alternative 5: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.8%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. lift-/.f32 N/A

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

   5. mult-flip N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. sub-negate-rev N/A

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

   14. lower--.f32 98.8%

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

10. Applied rewrites 98.8%

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

Alternative 6: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.8%

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

Alternative 7: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f32 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

8. Applied rewrites 99.0%

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

   1. lift-sin.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. metadata-eval N/A

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

   5. mult-flip N/A

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

   6. lift-PI.f32 N/A

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

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

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

   8. lift-neg.f32 N/A

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

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

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

   10. lift-+.f32 N/A

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

   11. distribute-lft-in N/A

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

   12. distribute-rgt-in N/A

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

   13. lift-+.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. cos-neg-rev N/A

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

   17. lift-cos.f32 98.9%

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

10. Applied rewrites 98.9%

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

Alternative 8: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.8%

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

4. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

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

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f32 98.9%

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

6. Applied rewrites 98.9%

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in maxCos around 0

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

    1. lower-+.f32 N/A

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

    2. lower-*.f32 98.1%

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

11. Applied rewrites 98.1%

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

Alternative 9: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-uy, \pi + \pi, \pi \cdot 0.5\right)\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} - 1\right) \cdot ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + maxCos, -ux, -2 \cdot \left(-ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= maxCos 4.0000000467443897e-7)
  (*
   (sin (fma (- uy) (+ PI PI) (* PI 0.5)))
   (sqrt (* (* (- (* 2.0 (/ 1.0 ux)) 1.0) ux) ux)))
  (*
   (cos (* (* uy 2.0) PI))
   (sqrt (fma (+ maxCos maxCos) (- ux) (* -2.0 (- ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = sinf(fmaf(-uy, (((float) M_PI) + ((float) M_PI)), (((float) M_PI) * 0.5f))) * sqrtf(((((2.0f * (1.0f / ux)) - 1.0f) * ux) * ux));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(fmaf((maxCos + maxCos), -ux, (-2.0f * -ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(sin(fma(Float32(-uy), Float32(Float32(pi) + Float32(pi)), Float32(Float32(pi) * Float32(0.5)))) * sqrt(Float32(Float32(Float32(Float32(Float32(2.0) * Float32(Float32(1.0) / ux)) - Float32(1.0)) * ux) * ux)));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(Float32(maxCos + maxCos), Float32(-ux), Float32(Float32(-2.0) * Float32(-ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 98.9%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f32 N/A

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\mathsf{fma}\left(-uy, \pi + \pi, \pi \cdot 0.5\right)\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} - 1\right) \cdot ux\right) \cdot ux} \]
   10. Step-by-step derivation

       1. lower--.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-/.f32 92.9%

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

   11. Applied rewrites 92.9%

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

   if 4.00000005e-7 < maxCos

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.7%

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f32 N/A

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

      5. distribute-rgt-neg-out N/A

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

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

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

      7. lift--.f32 N/A

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

      8. sub-flip N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. metadata-eval N/A

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

      17. lower-neg.f32 76.7%

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

   6. Applied rewrites 76.7%

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

Alternative 10: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{1}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(maxCos + maxCos, -ux, -2 \cdot \left(-ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= maxCos 4.0000000467443897e-7)
  (*
   (sqrt (* (- (* 2.0 (/ 1.0 ux)) 1.0) (* ux ux)))
   (cos (* (+ uy uy) PI)))
  (*
   (cos (* (* uy 2.0) PI))
   (sqrt (fma (+ maxCos maxCos) (- ux) (* -2.0 (- ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = sqrtf((((2.0f * (1.0f / ux)) - 1.0f) * (ux * ux))) * cosf(((uy + uy) * ((float) M_PI)));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(fmaf((maxCos + maxCos), -ux, (-2.0f * -ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(2.0) * Float32(Float32(1.0) / ux)) - Float32(1.0)) * Float32(ux * ux))) * cos(Float32(Float32(uy + uy) * Float32(pi))));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(Float32(maxCos + maxCos), Float32(-ux), Float32(Float32(-2.0) * Float32(-ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 98.9%

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

   6. Applied rewrites 98.9%

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower--.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-/.f32 92.7%

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

   11. Applied rewrites 92.7%

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

   if 4.00000005e-7 < maxCos

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.7%

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f32 N/A

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

      5. distribute-rgt-neg-out N/A

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

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

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

      7. lift--.f32 N/A

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

      8. sub-flip N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lower-neg.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. metadata-eval N/A

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

      17. lower-neg.f32 76.7%

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

   6. Applied rewrites 76.7%

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

Alternative 11: 89.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.0003600000054575503:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-uy, \pi + \pi, \pi \cdot 0.5\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.0003600000054575503)
  (*
   (sin (fma (- uy) (+ PI PI) (* PI 0.5)))
   (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
  (*
   (cos (* (* uy 2.0) PI))
   (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0003600000054575503f) {
		tmp = sinf(fmaf(-uy, (((float) M_PI) + ((float) M_PI)), (((float) M_PI) * 0.5f))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003600000054575503))
		tmp = Float32(sin(fma(Float32(-uy), Float32(Float32(pi) + Float32(pi)), Float32(Float32(pi) * Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.60000005e-4

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 98.9%

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\mathsf{fma}\left(-uy, \pi + \pi, \pi \cdot 0.5\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 - 2 \cdot maxCos\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.8%

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

   9. Applied rewrites 76.8%

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

   if 3.60000005e-4 < ux

   1. Initial program 57.6%

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

   2. Taylor expanded in maxCos around 0

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

      1. lower--.f32 55.8%

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 55.6%

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

   7. Applied rewrites 55.6%

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

Alternative 12: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.0003600000054575503:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.0003600000054575503)
  (*
   (sin (fma -2.0 (* uy PI) (* 0.5 PI)))
   (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
  (*
   (cos (* (* uy 2.0) PI))
   (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0003600000054575503f) {
		tmp = sinf(fmaf(-2.0f, (uy * ((float) M_PI)), (0.5f * ((float) M_PI)))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003600000054575503))
		tmp = Float32(sin(fma(Float32(-2.0), Float32(uy * Float32(pi)), Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.60000005e-4

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f32 98.9%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f32 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f32 N/A

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower-sin.f32 N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-PI.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. lower-PI.f32 N/A

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

       8. lower-sqrt.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lower--.f32 N/A

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

       11. lower-*.f32 76.7%

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

   11. Applied rewrites 76.7%

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

   if 3.60000005e-4 < ux

   1. Initial program 57.6%

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

   2. Taylor expanded in maxCos around 0

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

      1. lower--.f32 55.8%

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 55.6%

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

   7. Applied rewrites 55.6%

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

Alternative 13: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;ux \leq 0.0003600000054575503:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (cos (* (* uy 2.0) PI))))
  (if (<= ux 0.0003600000054575503)
    (* t_0 (sqrt (* ux (- (- 2.0 maxCos) maxCos))))
    (* t_0 (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (ux <= 0.0003600000054575503f) {
		tmp = t_0 * sqrtf((ux * ((2.0f - maxCos) - maxCos)));
	} else {
		tmp = t_0 * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003600000054575503))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))));
	else
		tmp = Float32(t_0 * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos(((uy * single(2.0)) * single(pi)));
	tmp = single(0.0);
	if (ux <= single(0.0003600000054575503))
		tmp = t_0 * sqrt((ux * ((single(2.0) - maxCos) - maxCos)));
	else
		tmp = t_0 * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.60000005e-4

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.7%

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

   4. Applied rewrites 76.7%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. associate--r+ N/A

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

      5. lower--.f32 N/A

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

      6. lower--.f32 76.7%

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

   6. Applied rewrites 76.7%

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

   if 3.60000005e-4 < ux

   1. Initial program 57.6%

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

   2. Taylor expanded in maxCos around 0

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

      1. lower--.f32 55.8%

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 55.6%

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

   7. Applied rewrites 55.6%

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

Alternative 14: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ t_2 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;t\_1 \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.02019999921321869:\\ \;\;\;\;t\_1 \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \mathsf{fma}\left(t\_2, 1, t\_2 \cdot \left(maxCos \cdot ux - ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos)))
       (t_1 (cos (* (* uy 2.0) PI)))
       (t_2 (- (fma maxCos ux 1.0) ux)))
  (if (<= (* t_1 (sqrt (- 1.0 (* t_0 t_0)))) 0.02019999921321869)
    (* t_1 (sqrt (* ux (- (- 2.0 maxCos) maxCos))))
    (sqrt (- 1.0 (fma t_2 1.0 (* t_2 (- (* maxCos ux) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float t_2 = fmaf(maxCos, ux, 1.0f) - ux;
	float tmp;
	if ((t_1 * sqrtf((1.0f - (t_0 * t_0)))) <= 0.02019999921321869f) {
		tmp = t_1 * sqrtf((ux * ((2.0f - maxCos) - maxCos)));
	} else {
		tmp = sqrtf((1.0f - fmaf(t_2, 1.0f, (t_2 * ((maxCos * ux) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	t_2 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	tmp = Float32(0.0)
	if (Float32(t_1 * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) <= Float32(0.02019999921321869))
		tmp = Float32(t_1 * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - fma(t_2, Float32(1.0), Float32(t_2 * Float32(Float32(maxCos * ux) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (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)))))) < 0.0201999992

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.7%

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

   4. Applied rewrites 76.7%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. associate--r+ N/A

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

      5. lower--.f32 N/A

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

      6. lower--.f32 76.7%

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

   6. Applied rewrites 76.7%

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

   if 0.0201999992 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (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 57.6%

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

   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. unpow2 N/A

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

      3. lift--.f32 N/A

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

      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. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+r- N/A

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

      9. lift--.f32 N/A

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

      10. lift-fma.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. associate--l+ N/A

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

      14. distribute-lft-in N/A

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

      15. lower-fma.f32 N/A

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

   6. Applied rewrites 49.8%

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

Alternative 15: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.02019999921321869:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \mathsf{fma}\left(t\_1, 1, t\_1 \cdot \left(maxCos \cdot ux - ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos)))
       (t_1 (- (fma maxCos ux 1.0) ux)))
  (if (<=
       (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))
       0.02019999921321869)
    (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) (cos (* (+ uy uy) PI)))
    (sqrt (- 1.0 (fma t_1 1.0 (* t_1 (- (* maxCos ux) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = fmaf(maxCos, ux, 1.0f) - ux;
	float tmp;
	if ((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) <= 0.02019999921321869f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * cosf(((uy + uy) * ((float) M_PI)));
	} else {
		tmp = sqrtf((1.0f - fmaf(t_1, 1.0f, (t_1 * ((maxCos * ux) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) <= Float32(0.02019999921321869))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * cos(Float32(Float32(uy + uy) * Float32(pi))));
	else
		tmp = sqrt(Float32(Float32(1.0) - fma(t_1, Float32(1.0), Float32(t_1 * Float32(Float32(maxCos * ux) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (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)))))) < 0.0201999992

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.7%

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

   4. Applied rewrites 76.7%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 76.7%

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

   6. Applied rewrites 76.7%

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

   if 0.0201999992 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (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 57.6%

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

   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. unpow2 N/A

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

      3. lift--.f32 N/A

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

      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. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+r- N/A

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

      9. lift--.f32 N/A

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

      10. lift-fma.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. associate--l+ N/A

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

      14. distribute-lft-in N/A

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

      15. lower-fma.f32 N/A

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

   6. Applied rewrites 49.8%

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

Alternative 16: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ t_2 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;t\_1 \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.013399999588727951:\\ \;\;\;\;t\_1 \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \mathsf{fma}\left(t\_2, 1, t\_2 \cdot \left(maxCos \cdot ux - ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos)))
       (t_1 (cos (* (* uy 2.0) PI)))
       (t_2 (- (fma maxCos ux 1.0) ux)))
  (if (<= (* t_1 (sqrt (- 1.0 (* t_0 t_0)))) 0.013399999588727951)
    (* t_1 (sqrt (* ux 2.0)))
    (sqrt (- 1.0 (fma t_2 1.0 (* t_2 (- (* maxCos ux) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float t_2 = fmaf(maxCos, ux, 1.0f) - ux;
	float tmp;
	if ((t_1 * sqrtf((1.0f - (t_0 * t_0)))) <= 0.013399999588727951f) {
		tmp = t_1 * sqrtf((ux * 2.0f));
	} else {
		tmp = sqrtf((1.0f - fmaf(t_2, 1.0f, (t_2 * ((maxCos * ux) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	t_2 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	tmp = Float32(0.0)
	if (Float32(t_1 * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) <= Float32(0.013399999588727951))
		tmp = Float32(t_1 * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = sqrt(Float32(Float32(1.0) - fma(t_2, Float32(1.0), Float32(t_2 * Float32(Float32(maxCos * ux) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (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)))))) < 0.0133999996

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.7%

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in maxCos around 0

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

   7. Applied rewrites 73.1%

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

   if 0.0133999996 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (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 57.6%

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

   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. unpow2 N/A

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

      3. lift--.f32 N/A

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

      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. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+r- N/A

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

      9. lift--.f32 N/A

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

      10. lift-fma.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-+.f32 N/A

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

      13. associate--l+ N/A

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

      14. distribute-lft-in N/A

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

      15. lower-fma.f32 N/A

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

   6. Applied rewrites 49.8%

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

Alternative 17: 51.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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. lift-pow.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-pow.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f32 N/A

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

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

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

   10. unpow2 N/A

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

   11. lower-fma.f32 N/A

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

6. Applied rewrites 51.2%

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

Alternative 18: 49.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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. unpow2 N/A

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

   3. lift--.f32 N/A

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

   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. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-+r- N/A

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

   9. lift--.f32 N/A

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

   10. lift-fma.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. associate--l+ N/A

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

   14. distribute-lft-in N/A

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

   15. lower-fma.f32 N/A

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

6. Applied rewrites 49.8%

   \[\leadsto \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, 1, \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(maxCos \cdot ux - ux\right)\right)} \]
7. Add Preprocessing

Alternative 19: 49.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - ((1.0e0 - (ux - (maxcos * ux))) ** 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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-+.f32 N/A

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

   3. associate--l+ N/A

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

   4. add-flip N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. sub-negate-rev N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 49.4%

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

6. Applied rewrites 49.4%

   \[\leadsto \sqrt{1 - {\left(1 - \left(ux - maxCos \cdot ux\right)\right)}^{2}} \]
7. Add Preprocessing

Alternative 20: 49.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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-+.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. lift-fma.f32 49.4%

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

6. Applied rewrites 49.4%

   \[\leadsto \sqrt{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}} \]
7. Add Preprocessing

Alternative 21: 49.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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. lift-+.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. associate-+r- N/A

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

   8. lift--.f32 N/A

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

   9. lift-fma.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 49.4%

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

   12. lift-fma.f32 N/A

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

   13. lift--.f32 N/A

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

   14. associate-+r- N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-+.f32 N/A

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

   19. lift--.f32 49.3%

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

   20. lift-+.f32 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. +-commutative N/A

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

   24. *-commutative N/A

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

   25. lower-fma.f32 49.3%

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

6. Applied rewrites 49.3%

   \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]
7. Add Preprocessing

Alternative 22: 49.3% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (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) {
	return sqrtf(fmaf((fmaf(maxCos, ux, 1.0f) - ux), (ux - fmaf(maxCos, ux, 1.0f)), 1.0f));
}

Julia

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

Derivation

1. Initial program 57.6%

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

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. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-+r- N/A

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

   9. lift--.f32 N/A

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

   10. +-commutative N/A

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

   11. lift--.f32 N/A

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

   12. pow2 N/A

       \[\leadsto \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   13. fp-cancel-sub-sign-inv N/A

       \[\leadsto \sqrt{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   14. lift--.f32 N/A

       \[\leadsto \sqrt{1 + \left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   15. +-commutative N/A

       \[\leadsto \sqrt{1 + \left(\mathsf{neg}\left(\left(ux \cdot maxCos + \left(1 - ux\right)\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   16. lift-fma.f32 N/A

       \[\leadsto \sqrt{1 + \left(\mathsf{neg}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

6. Applied rewrites 49.3%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)} \]
7. Add Preprocessing

Alternative 23: 40.6% accurate, 4.4× speedup

Math

\[\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, 2, -2\right), ux, 1\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (sqrt (- 1.0 (fma (fma maxCos 2.0 -2.0) ux 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((1.0f - fmaf(fmaf(maxCos, 2.0f, -2.0f), ux, 1.0f)));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(1.0) - fma(fma(maxCos, Float32(2.0), Float32(-2.0)), ux, Float32(1.0))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

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

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. lower-*.f32 40.6%

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied rewrites 40.6%

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]
8. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(ux \cdot \left(2 \cdot maxCos - 2\right) + 1\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(ux \cdot \left(2 \cdot maxCos - 2\right) + 1\right)} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{1 - \left(\left(2 \cdot maxCos - 2\right) \cdot ux + 1\right)} \]

   5. lower-fma.f32 40.6%

      \[\leadsto \sqrt{1 - \mathsf{fma}\left(2 \cdot maxCos - 2, ux, 1\right)} \]

   6. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \mathsf{fma}\left(2 \cdot maxCos - 2, ux, 1\right)} \]

   7. sub-flip N/A

      \[\leadsto \sqrt{1 - \mathsf{fma}\left(2 \cdot maxCos + \left(\mathsf{neg}\left(2\right)\right), ux, 1\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \sqrt{1 - \mathsf{fma}\left(2 \cdot maxCos + \left(\mathsf{neg}\left(2\right)\right), ux, 1\right)} \]

   9. *-commutative N/A

      \[\leadsto \sqrt{1 - \mathsf{fma}\left(maxCos \cdot 2 + \left(\mathsf{neg}\left(2\right)\right), ux, 1\right)} \]

   10. lower-fma.f32 N/A

       \[\leadsto \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, 2, \mathsf{neg}\left(2\right)\right), ux, 1\right)} \]

   11. metadata-eval 40.6%

       \[\leadsto \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, 2, -2\right), ux, 1\right)} \]

9. Applied rewrites 40.6%

   \[\leadsto \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, 2, -2\right), ux, 1\right)} \]
10. Add Preprocessing

Alternative 24: 39.9% accurate, 6.1× speedup

Math

\[\sqrt{1 - \left(1 + ux \cdot -2\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (sqrt (- 1.0 (+ 1.0 (* ux -2.0)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((1.0f - (1.0f + (ux * -2.0f))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - (1.0e0 + (ux * (-2.0e0)))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(1.0) - Float32(Float32(1.0) + Float32(ux * Float32(-2.0)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(1.0) - (single(1.0) + (ux * single(-2.0)))));
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

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

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. lower-*.f32 40.6%

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied rewrites 40.6%

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot -2\right)} \]
9. Step-by-step derivation

10. Applied rewrites 39.9%

    \[\leadsto \sqrt{1 - \left(1 + ux \cdot -2\right)} \]
11. Add Preprocessing

Alternative 25: 6.6% accurate, 22.7× speedup

Math

\[\sqrt{0} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (sqrt 0.0))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(0.0f);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(0.0e0)
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(0.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(single(0.0));
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

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

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]
6. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   3. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

   4. lower-*.f32 40.6%

      \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

7. Applied rewrites 40.6%

   \[\leadsto \sqrt{1 - \left(1 + ux \cdot \left(2 \cdot maxCos - 2\right)\right)} \]

8. Taylor expanded in ux around 0

   \[\leadsto \sqrt{1 - 1} \]
9. Step-by-step derivation

10. Applied rewrites 6.6%

    \[\leadsto \sqrt{1 - 1} \]

11. Evaluated real constant 6.6%

    \[\leadsto \sqrt{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))))))