UniformSampleCone, x

Percentage Accurate: 57.7% → 99.0%

Time: 8.1s

Alternatives: 22

Speedup: 11.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 98.9%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. count-2-rev N/A

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

   11. lower-+.f32 98.3

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

9. Applied rewrites 98.3%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower--.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lift-neg.f32 98.3

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

9. Applied rewrites 98.3%

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

Alternative 5: 97.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 97.5

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

7. Applied rewrites 97.5%

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

Alternative 6: 97.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 97.5

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

9. Applied rewrites 97.5%

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

Alternative 7: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9955000281333923:\\ \;\;\;\;\sqrt{\left(\left(\left(-ux\right) + 2\right) - maxCos\right) \cdot ux} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.9955000281333923)
   (* (sqrt (* (- (+ (- ux) 2.0) maxCos) ux)) (cos (* (+ uy uy) PI)))
   (*
    (sqrt
     (*
      (- (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) maxCos) maxCos)
      ux))
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9955000281333923f) {
		tmp = sqrtf((((-ux + 2.0f) - maxCos) * ux)) * cosf(((uy + uy) * ((float) M_PI)));
	} else {
		tmp = sqrtf((((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - maxCos) - maxCos) * ux)) * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9955000281333923))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(-ux) + Float32(2.0)) - maxCos) * ux)) * cos(Float32(Float32(uy + uy) * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * ux)) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.995500028

   1. Initial program 57.9%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 97.4

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

   4. Applied rewrites 97.4%

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

   6. Applied rewrites 97.4%

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

   7. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f32 92.0

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

   9. Applied rewrites 92.0%

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

   if 0.995500028 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 98.8

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

   9. Applied rewrites 98.8%

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

Alternative 8: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.014999999664723873)
   (*
    (sqrt
     (*
      (- (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) maxCos) maxCos)
      ux))
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))
   (* (sqrt (* (+ (- ux) 2.0) ux)) (cos (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.014999999664723873f) {
		tmp = sqrtf((((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - maxCos) - maxCos) * ux)) * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf(((-ux + 2.0f) * ux)) * cosf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.014999999664723873))
		tmp = Float32(sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * ux)) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(-ux) + Float32(2.0)) * ux)) * cos(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0149999997

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 98.8

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

   9. Applied rewrites 98.8%

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

   if 0.0149999997 < uy

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f32 91.5

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

   7. Applied rewrites 91.5%

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

Alternative 9: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9660000205039978:\\ \;\;\;\;\sqrt{ux + ux} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.9660000205039978)
   (* (sqrt (+ ux ux)) (cos (* PI (+ uy uy))))
   (*
    (sqrt
     (*
      (- (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) maxCos) maxCos)
      ux))
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9660000205039978f) {
		tmp = sqrtf((ux + ux)) * cosf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = sqrtf((((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - maxCos) - maxCos) * ux)) * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9660000205039978))
		tmp = Float32(sqrt(Float32(ux + ux)) * cos(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * ux)) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.966000021

   1. Initial program 58.0%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 72.4

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

   7. Applied rewrites 72.4%

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

   if 0.966000021 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.3

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

   4. Applied rewrites 99.3%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 97.4

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

   9. Applied rewrites 97.4%

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

Alternative 10: 88.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 99.0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 88.2

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

9. Applied rewrites 88.2%

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

Alternative 11: 83.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0009200000204145908:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{2}\right)\right), \sqrt{ux \cdot 2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0009200000204145908)
   (*
    1.0
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (fma
    -2.0
    (* (sqrt ux) (* (* uy uy) (* (* PI PI) (sqrt 2.0))))
    (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0009200000204145908f) {
		tmp = 1.0f * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = fmaf(-2.0f, (sqrtf(ux) * ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(2.0f)))), sqrtf((ux * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0009200000204145908))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = fma(Float32(-2.0), Float32(sqrt(ux) * Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(2.0))))), sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.2000002e-4

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in uy around 0

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

   7. Applied rewrites 97.2%

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

   if 9.2000002e-4 < uy

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 72.6

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-sqrt.f32 N/A

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

      13. sqrt-unprod N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lower-*.f32 53.1

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

   10. Applied rewrites 53.1%

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

Alternative 12: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.999983012676239:\\ \;\;\;\;\left(\sqrt{2} + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.999983012676239)
   (* (+ (sqrt 2.0) (* -2.0 (* (* uy uy) (* (* PI PI) (sqrt 2.0))))) (sqrt ux))
   (*
    1.0
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.999983012676239f) {
		tmp = (sqrtf(2.0f) + (-2.0f * ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(2.0f))))) * sqrtf(ux);
	} else {
		tmp = 1.0f * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.999983012676239))
		tmp = Float32(Float32(sqrt(Float32(2.0)) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(2.0)))))) * sqrt(ux));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999983013

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 72.6

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-sqrt.f32 53.1

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

   10. Applied rewrites 53.1%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in uy around 0

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

   7. Applied rewrites 97.2%

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

Alternative 13: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.999983012676239:\\ \;\;\;\;\left(\sqrt{2} + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.999983012676239)
   (* (+ (sqrt 2.0) (* -2.0 (* (* uy uy) (* (* PI PI) (sqrt 2.0))))) (sqrt ux))
   (*
    (sqrt
     (*
      (- (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) maxCos) maxCos)
      ux))
    1.0)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.999983012676239f) {
		tmp = (sqrtf(2.0f) + (-2.0f * ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(2.0f))))) * sqrtf(ux);
	} else {
		tmp = sqrtf((((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - maxCos) - maxCos) * ux)) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.999983012676239))
		tmp = Float32(Float32(sqrt(Float32(2.0)) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(2.0)))))) * sqrt(ux));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * ux)) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999983013

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 72.6

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-sqrt.f32 53.1

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

   10. Applied rewrites 53.1%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.4

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

   4. Applied rewrites 99.4%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in uy around 0

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

   9. Applied rewrites 97.2%

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

Alternative 14: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.999983012676239:\\ \;\;\;\;\left(\sqrt{2} + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(\frac{-2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.999983012676239)
   (* (+ (sqrt 2.0) (* -2.0 (* (* uy uy) (* (* PI PI) (sqrt 2.0))))) (sqrt ux))
   (* 1.0 (sqrt (* (- (fma (/ -2.0 ux) maxCos (/ 2.0 ux)) 1.0) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.999983012676239f) {
		tmp = (sqrtf(2.0f) + (-2.0f * ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * sqrtf(2.0f))))) * sqrtf(ux);
	} else {
		tmp = 1.0f * sqrtf(((fmaf((-2.0f / ux), maxCos, (2.0f / ux)) - 1.0f) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.999983012676239))
		tmp = Float32(Float32(sqrt(Float32(2.0)) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(Float32(2.0)))))) * sqrt(ux));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(fma(Float32(Float32(-2.0) / ux), maxCos, Float32(Float32(2.0) / ux)) - Float32(1.0)) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999983013

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 72.6

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-sqrt.f32 53.1

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

   10. Applied rewrites 53.1%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower--.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 97.0%

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

   11. Taylor expanded in ux around 0

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

       1. lower-/.f32 95.5

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

   13. Applied rewrites 95.5%

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

Alternative 15: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.999983012676239:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(\frac{-2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.999983012676239)
   (* (* (sqrt 2.0) (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))) (sqrt ux))
   (* 1.0 (sqrt (* (- (fma (/ -2.0 ux) maxCos (/ 2.0 ux)) 1.0) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.999983012676239f) {
		tmp = (sqrtf(2.0f) * (1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))))) * sqrtf(ux);
	} else {
		tmp = 1.0f * sqrtf(((fmaf((-2.0f / ux), maxCos, (2.0f / ux)) - 1.0f) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.999983012676239))
		tmp = Float32(Float32(sqrt(Float32(2.0)) * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))))) * sqrt(ux));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(fma(Float32(Float32(-2.0) / ux), maxCos, Float32(Float32(2.0) / ux)) - Float32(1.0)) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999983013

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 72.6

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 53.1

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

   10. Applied rewrites 53.1%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower--.f32 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f32 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 97.0%

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

   11. Taylor expanded in ux around 0

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

       1. lower-/.f32 95.5

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

   13. Applied rewrites 95.5%

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

Alternative 16: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.999983012676239:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* uy 2.0) PI)) 0.999983012676239)
   (* (* (sqrt 2.0) (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))) (sqrt ux))
   (* 1.0 (sqrt (* (- (/ (+ 2.0 (* -2.0 maxCos)) ux) 1.0) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.999983012676239f) {
		tmp = (sqrtf(2.0f) * (1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))))) * sqrtf(ux);
	} else {
		tmp = 1.0f * sqrtf(((((2.0f + (-2.0f * maxCos)) / ux) - 1.0f) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.999983012676239))
		tmp = Float32(Float32(sqrt(Float32(2.0)) * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))))) * sqrt(ux));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos)) / ux) - Float32(1.0)) * Float32(ux * ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos(((uy * single(2.0)) * single(pi))) <= single(0.999983012676239))
		tmp = (sqrt(single(2.0)) * (single(1.0) + (single(-2.0) * ((uy * uy) * (single(pi) * single(pi)))))) * sqrt(ux);
	else
		tmp = single(1.0) * sqrt(((((single(2.0) + (single(-2.0) * maxCos)) / ux) - single(1.0)) * (ux * ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999983013

   1. Initial program 57.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 56.1

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-cos.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 72.6

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

   7. Applied rewrites 72.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 53.1

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

   10. Applied rewrites 53.1%

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

   if 0.999983013 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. Taylor expanded in ux around -inf

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right) \cdot \color{blue}{{ux}^{2}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right) \cdot \color{blue}{{ux}^{2}}} \]

   4. Applied rewrites 99.2%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(\left(-maxCos\right) + 1\right) \cdot \left(\left(-maxCos\right) + 1\right)\right) \cdot \left(ux \cdot ux\right)}} \]

   5. Taylor expanded in maxCos around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} + maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right)\right) - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]
   6. Step-by-step derivation

      1. lower--.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} + maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right)\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      2. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right) + 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      3. *-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right) \cdot maxCos + 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      4. lower-fma.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      5. lower--.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      6. mul-1-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      7. lift-neg.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + \left(-maxCos\right)\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      8. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      9. lower-+.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      10. associate-*r/ N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2 \cdot 1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      11. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      12. lower-/.f32 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2 \cdot 1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

      15. lower-/.f32 99.2

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   7. Applied rewrites 99.2%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{1} \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.0%

       \[\leadsto \color{blue}{1} \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   11. Taylor expanded in ux around 0

       \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

       2. lower-+.f32 N/A

          \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

       3. lower-*.f32 95.6

          \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

   13. Applied rewrites 95.6%

       \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 78.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* 1.0 (sqrt (* (- (/ (+ 2.0 (* -2.0 maxCos)) ux) 1.0) (* ux ux)))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f * sqrtf(((((2.0f + (-2.0f * maxCos)) / ux) - 1.0f) * (ux * ux)));
}

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 = 1.0e0 * sqrt(((((2.0e0 + ((-2.0e0) * maxcos)) / ux) - 1.0e0) * (ux * ux)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(Float32(Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos)) / ux) - Float32(1.0)) * Float32(ux * ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) * sqrt(((((single(2.0) + (single(-2.0) * maxCos)) / ux) - single(1.0)) * (ux * ux)));
end

Derivation

1. Initial program 57.7%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in ux around -inf

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right) \cdot \color{blue}{{ux}^{2}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right) \cdot \color{blue}{{ux}^{2}}} \]

4. Applied rewrites 98.8%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(\left(-maxCos\right) + 1\right) \cdot \left(\left(-maxCos\right) + 1\right)\right) \cdot \left(ux \cdot ux\right)}} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} + maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right)\right) - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]
6. Step-by-step derivation

   1. lower--.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} + maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right)\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   2. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right) + 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   3. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right) \cdot maxCos + 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   4. lower-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   5. lower--.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   7. lift-neg.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + \left(-maxCos\right)\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   8. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   9. lower-+.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   10. associate-*r/ N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2 \cdot 1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   11. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   12. lower-/.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   13. associate-*r/ N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2 \cdot 1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   15. lower-/.f32 98.8

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

7. Applied rewrites 98.8%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]
9. Step-by-step derivation

10. Applied rewrites 79.5%

    \[\leadsto \color{blue}{1} \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

11. Taylor expanded in ux around 0

    \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

    2. lower-+.f32 N/A

       \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

    3. lower-*.f32 78.5

       \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

13. Applied rewrites 78.5%

    \[\leadsto 1 \cdot \sqrt{\left(\frac{2 + -2 \cdot maxCos}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
14. Add Preprocessing

Alternative 18: 75.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* 1.0 (sqrt (* (- (/ 2.0 ux) 1.0) (* ux ux)))))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f * sqrtf((((2.0f / ux) - 1.0f) * (ux * ux)));
}

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 = 1.0e0 * sqrt((((2.0e0 / ux) - 1.0e0) * (ux * ux)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(Float32(Float32(2.0) / ux) - Float32(1.0)) * Float32(ux * ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) * sqrt((((single(2.0) / ux) - single(1.0)) * (ux * ux)));
end

Derivation

1. Initial program 57.7%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in ux around -inf

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right) \cdot \color{blue}{{ux}^{2}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right) - {\left(1 + -1 \cdot maxCos\right)}^{2}\right) \cdot \color{blue}{{ux}^{2}}} \]

4. Applied rewrites 98.8%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(\left(-maxCos\right) + 1\right) \cdot \left(\left(-maxCos\right) + 1\right)\right) \cdot \left(ux \cdot ux\right)}} \]

5. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} + maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right)\right) - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]
6. Step-by-step derivation

   1. lower--.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(2 \cdot \frac{1}{ux} + maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right)\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   2. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(maxCos \cdot \left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right) + 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   3. *-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}\right) \cdot maxCos + 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   4. lower-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   5. lower--.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + -1 \cdot maxCos\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   7. lift-neg.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(2 + \left(-maxCos\right)\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   8. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   9. lower-+.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - 2 \cdot \frac{1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   10. associate-*r/ N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2 \cdot 1}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   11. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   12. lower-/.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, 2 \cdot \frac{1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   13. associate-*r/ N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2 \cdot 1}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

   15. lower-/.f32 98.8

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

7. Applied rewrites 98.8%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{1} \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]
9. Step-by-step derivation

10. Applied rewrites 79.5%

    \[\leadsto \color{blue}{1} \cdot \sqrt{\left(\mathsf{fma}\left(\left(\left(-maxCos\right) + 2\right) - \frac{2}{ux}, maxCos, \frac{2}{ux}\right) - 1\right) \cdot \left(ux \cdot ux\right)} \]

11. Taylor expanded in maxCos around 0

    \[\leadsto 1 \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
12. Step-by-step derivation

    1. lift-/.f32 75.0

       \[\leadsto 1 \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

13. Applied rewrites 75.0%

    \[\leadsto 1 \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
14. Add Preprocessing

Alternative 19: 73.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{0.5 \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (sqrt (* 0.5 (* ux (- 4.0 (* 4.0 maxCos)))))
   (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = sqrtf((0.5f * (ux * (4.0f - (4.0f * maxCos)))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

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
    real(4) :: tmp
    if (ux <= 0.00019999999494757503e0) then
        tmp = sqrt((0.5e0 * (ux * (4.0e0 - (4.0e0 * maxcos)))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) * (1.0e0 - ux))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = sqrt(Float32(Float32(0.5) * Float32(ux * Float32(Float32(4.0) - Float32(Float32(4.0) * maxCos)))));
	else
		tmp = 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)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = sqrt((single(0.5) * (ux * (single(4.0) - (single(4.0) * maxCos)))));
	else
		tmp = 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 < 1.99999995e-4

   1. Initial program 37.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   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. unpow2 N/A

         \[\leadsto \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      4. lower-*.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. lower--.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

         \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)} \]

      10. lower-fma.f32 34.0

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

   4. Applied rewrites 34.0%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]
   5. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

      3. lift--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

      4. lift-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

      5. lift--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

      6. lift-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)} \]

      7. flip-- N/A

         \[\leadsto \sqrt{\frac{1 \cdot 1 - \left(\left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)\right) \cdot \left(\left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)\right)}{1 + \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)}} \]

      8. lower-/.f32 N/A

         \[\leadsto \sqrt{\frac{1 \cdot 1 - \left(\left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)\right) \cdot \left(\left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)\right)}{1 + \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)}} \]

   6. Applied rewrites 34.0%

      \[\leadsto \sqrt{\frac{1 - \left(\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)\right) \cdot \left(\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)\right)}{1 + \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   7. Taylor expanded in ux around 0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)} \]

      3. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)} \]

      4. lower-*.f32 75.5

         \[\leadsto \sqrt{0.5 \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)} \]

   9. Applied rewrites 75.5%

      \[\leadsto \sqrt{0.5 \cdot \left(ux \cdot \left(4 - 4 \cdot maxCos\right)\right)} \]

   if 1.99999995e-4 < ux

   1. Initial program 89.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   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. unpow2 N/A

         \[\leadsto \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      4. lower-*.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. lower--.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

         \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)} \]

      10. lower-fma.f32 73.4

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

   4. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.7%

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.4%

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 71.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9995999932289124)
     (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))
     (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9995999932289124f) {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	} else {
		tmp = sqrtf((ux * 2.0f));
	}
	return tmp;
}

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (1.0e0 - ux) + (ux * maxcos)
    if ((t_0 * t_0) <= 0.9995999932289124e0) then
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) * (1.0e0 - ux))))
    else
        tmp = sqrt((ux * 2.0e0))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9995999932289124))
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if ((t_0 * t_0) <= single(0.9995999932289124))
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	else
		tmp = sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999599993

   1. Initial program 89.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   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. unpow2 N/A

         \[\leadsto \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      4. lower-*.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. lower--.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

         \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      8. lower--.f32 N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)} \]

      10. lower-fma.f32 73.4

          \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

   4. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.8%

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]

   if 0.999599993 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 37.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \cos \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      7. lift--.f32 N/A

         \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      8. lift--.f32 N/A

         \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

      11. lower-cos.f32 N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

      13. lower-*.f32 N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

      14. lift-PI.f32 N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \]

      16. count-2-rev N/A

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right) \]

      17. lower-+.f32 36.9

          \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right) \]

   4. Applied rewrites 36.9%

      \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \sqrt{ux} \cdot \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

      6. associate-*r* N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

      8. lift-*.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

      9. lift-*.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

      10. lift-PI.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

      11. lift-cos.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

      12. lift-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

      14. count-2-rev N/A

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

      15. lift-+.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

      16. lower-sqrt.f32 86.5

          \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   7. Applied rewrites 86.5%

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux}} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \sqrt{ux} \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto \sqrt{ux \cdot 2} \]

      2. lower-sqrt.f32 N/A

         \[\leadsto \sqrt{ux \cdot 2} \]

      3. lower-*.f32 71.4

         \[\leadsto \sqrt{ux \cdot 2} \]

   10. Applied rewrites 71.4%

       \[\leadsto \sqrt{ux \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 61.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 2} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux 2.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((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((ux * 2.0e0))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(2.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * single(2.0)));
end

Derivation

1. Initial program 57.7%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \cos \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   11. lower-cos.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \]

   16. count-2-rev N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right) \]

   17. lower-+.f32 55.7

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right) \]

4. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \cos \left(\pi \cdot \left(uy + uy\right)\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux} \cdot \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   6. associate-*r* N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   7. *-commutative N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   8. lift-*.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   9. lift-*.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   10. lift-PI.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   11. lift-cos.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   12. lift-*.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   13. *-commutative N/A

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(2 \cdot uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   14. count-2-rev N/A

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   15. lift-+.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   16. lower-sqrt.f32 72.7

       \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

7. Applied rewrites 72.7%

   \[\leadsto \left(\sqrt{2} \cdot \cos \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux}} \]

8. Taylor expanded in uy around 0

   \[\leadsto \sqrt{ux} \cdot \sqrt{2} \]
9. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \sqrt{ux \cdot 2} \]

   2. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{ux \cdot 2} \]

   3. lower-*.f32 61.5

      \[\leadsto \sqrt{ux \cdot 2} \]

10. Applied rewrites 61.5%

    \[\leadsto \sqrt{ux \cdot 2} \]
11. Add Preprocessing

Alternative 22: 6.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (- 1.0 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((1.0f - 1.0f));
}

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))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(1.0) - Float32(1.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 57.7%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. 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. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   4. lower-*.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. lower--.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

      \[\leadsto \sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   9. +-commutative N/A

      \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\left(maxCos \cdot ux + 1\right) - ux\right)} \]

   10. lower-fma.f32 49.4

       \[\leadsto \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \]

4. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{1 - 1} \]
6. Step-by-step derivation

7. Applied rewrites 6.6%

   \[\leadsto \sqrt{1 - 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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)))))))