UniformSampleCone, x

Percentage Accurate: 57.5% → 99.0%

Time: 5.2s

Alternatives: 18

Speedup: 1.8×

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: 57.5% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-pow.f32 N/A

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

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

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

   7. mul-1-neg N/A

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

   8. lift-fma.f32 N/A

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

   9. associate-+r+ N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-in N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-*.f32 N/A

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

8. Applied rewrites 99.0%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 98.9%

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

Alternative 3: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f32 98.2

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

7. Applied rewrites 98.2%

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

Alternative 4: 97.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 97.4

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

7. Applied rewrites 97.4%

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

Alternative 5: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 4.099999932805076 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-ux, {\left(maxCos - 1\right)}^{2}, \mathsf{fma}\left(-2, maxCos, 2\right)\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2, \left(-1 \cdot ux\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 4.099999932805076e-5)
   (*
    (sqrt (* (fma (- ux) (pow (- maxCos 1.0) 2.0) (fma -2.0 maxCos 2.0)) ux))
    1.0)
   (* (sqrt (fma ux 2.0 (* (* -1.0 ux) ux))) (cos (* PI (* 2.0 uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 4.099999932805076e-5f) {
		tmp = sqrtf((fmaf(-ux, powf((maxCos - 1.0f), 2.0f), fmaf(-2.0f, maxCos, 2.0f)) * ux)) * 1.0f;
	} else {
		tmp = sqrtf(fmaf(ux, 2.0f, ((-1.0f * ux) * ux))) * cosf((((float) M_PI) * (2.0f * uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(4.099999932805076e-5))
		tmp = Float32(sqrt(Float32(fma(Float32(-ux), (Float32(maxCos - Float32(1.0)) ^ Float32(2.0)), fma(Float32(-2.0), maxCos, Float32(2.0))) * ux)) * Float32(1.0));
	else
		tmp = Float32(sqrt(fma(ux, Float32(2.0), Float32(Float32(Float32(-1.0) * ux) * ux))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 4.09999993e-5

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative 99.5

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

      2. *-commutative 99.5

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

   9. Applied rewrites 99.5%

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

   if 4.09999993e-5 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.2

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

   4. Applied rewrites 98.2%

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

   6. Applied rewrites 98.2%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-pow.f32 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-fma.f32 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower-*.f32 N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 92.4

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

   11. Applied rewrites 92.4%

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

Alternative 6: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;maxCos \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2, \left(-1 \cdot ux\right) \cdot ux\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2, \left(-2 \cdot maxCos\right) \cdot ux\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* 2.0 uy)))))
   (if (<= maxCos 0.00019999999494757503)
     (* (sqrt (fma ux 2.0 (* (* -1.0 ux) ux))) t_0)
     (* (sqrt (fma ux 2.0 (* (* -2.0 maxCos) ux))) t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (2.0f * uy)));
	float tmp;
	if (maxCos <= 0.00019999999494757503f) {
		tmp = sqrtf(fmaf(ux, 2.0f, ((-1.0f * ux) * ux))) * t_0;
	} else {
		tmp = sqrtf(fmaf(ux, 2.0f, ((-2.0f * maxCos) * ux))) * t_0;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00019999999494757503))
		tmp = Float32(sqrt(fma(ux, Float32(2.0), Float32(Float32(Float32(-1.0) * ux) * ux))) * t_0);
	else
		tmp = Float32(sqrt(fma(ux, Float32(2.0), Float32(Float32(Float32(-2.0) * maxCos) * ux))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-4

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.9

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-pow.f32 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-fma.f32 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower-*.f32 N/A

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 97.6

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

   11. Applied rewrites 97.6%

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

   if 1.99999995e-4 < maxCos

   1. Initial program 56.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-pow.f32 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-fma.f32 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower-*.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in ux around 0

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

       1. lift-*.f32 76.9

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

   11. Applied rewrites 76.9%

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

Alternative 7: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 0.00019999999494757503:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(-1, ux, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2, \left(-2 \cdot maxCos\right) \cdot ux\right)} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 0.00019999999494757503)
   (* (cos (* (* uy 2.0) PI)) (sqrt (* (fma -1.0 ux 2.0) ux)))
   (* (sqrt (fma ux 2.0 (* (* -2.0 maxCos) ux))) (cos (* PI (* 2.0 uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 0.00019999999494757503f) {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((fmaf(-1.0f, ux, 2.0f) * ux));
	} else {
		tmp = sqrtf(fmaf(ux, 2.0f, ((-2.0f * maxCos) * ux))) * cosf((((float) M_PI) * (2.0f * uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00019999999494757503))
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(fma(Float32(-1.0), ux, Float32(2.0)) * ux)));
	else
		tmp = Float32(sqrt(fma(ux, Float32(2.0), Float32(Float32(Float32(-2.0) * maxCos) * ux))) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-4

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 97.5

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

   7. Applied rewrites 97.5%

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

   if 1.99999995e-4 < maxCos

   1. Initial program 56.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.9%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-pow.f32 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lift-fma.f32 N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-in N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower-*.f32 N/A

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in ux around 0

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

       1. lift-*.f32 76.9

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

   11. Applied rewrites 76.9%

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

Alternative 8: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;maxCos \leq 0.00019999999494757503:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-1, ux, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (* uy 2.0) PI))))
   (if (<= maxCos 0.00019999999494757503)
     (* t_0 (sqrt (* (fma -1.0 ux 2.0) ux)))
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (maxCos <= 0.00019999999494757503f) {
		tmp = t_0 * sqrtf((fmaf(-1.0f, ux, 2.0f) * ux));
	} else {
		tmp = t_0 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00019999999494757503))
		tmp = Float32(t_0 * sqrt(Float32(fma(Float32(-1.0), ux, Float32(2.0)) * ux)));
	else
		tmp = Float32(t_0 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-4

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 97.5

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

   7. Applied rewrites 97.5%

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

   if 1.99999995e-4 < maxCos

   1. Initial program 56.3%

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

   2. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.9

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

   4. Applied rewrites 76.9%

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

Alternative 9: 92.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 92.6

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

7. Applied rewrites 92.6%

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

Alternative 10: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - e^{\mathsf{log1p}\left(maxCos \cdot ux - ux\right) \cdot 2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))
        0.017999999225139618)
     (* (sqrt (* 2.0 ux)) (cos (* PI (* 2.0 uy))))
     (sqrt (- 1.0 (exp (* (log1p (- (* maxCos ux) ux)) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) <= 0.017999999225139618f) {
		tmp = sqrtf((2.0f * ux)) * cosf((((float) M_PI) * (2.0f * uy)));
	} else {
		tmp = sqrtf((1.0f - expf((log1pf(((maxCos * ux) - ux)) * 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) <= Float32(0.017999999225139618))
		tmp = Float32(sqrt(Float32(Float32(2.0) * ux)) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	else
		tmp = sqrt(Float32(Float32(1.0) - exp(Float32(log1p(Float32(Float32(maxCos * ux) - ux)) * Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0179999992

   1. Initial program 38.5%

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

   2. Taylor expanded in 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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. lift-fma.f32 91.3

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

   9. Applied rewrites 91.3%

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

   10. Taylor expanded in maxCos around 0

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

   12. Applied rewrites 86.1%

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

   if 0.0179999992 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 89.2%

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

   2. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 78.0

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

   4. Applied rewrites 78.0%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. associate--l+ N/A

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

      9. lower-log1p.f32 N/A

         \[\leadsto \sqrt{1 - e^{\mathsf{log1p}\left(maxCos \cdot ux - ux\right) \cdot 2}} \]

      10. lower--.f32 N/A

          \[\leadsto \sqrt{1 - e^{\mathsf{log1p}\left(maxCos \cdot ux - ux\right) \cdot 2}} \]

      11. lower-*.f32 79.6

          \[\leadsto \sqrt{1 - e^{\mathsf{log1p}\left(maxCos \cdot ux - ux\right) \cdot 2}} \]

   6. Applied rewrites 79.6%

      \[\leadsto \sqrt{1 - e^{\mathsf{log1p}\left(maxCos \cdot ux - ux\right) \cdot 2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \sin^{-1} \left(1 + ux \cdot \left(maxCos - 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))
        0.019999999552965164)
     (* (sqrt (* 2.0 ux)) (cos (* PI (* 2.0 uy))))
     (cos (asin (+ 1.0 (* ux (- maxCos 1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) <= 0.019999999552965164f) {
		tmp = sqrtf((2.0f * ux)) * cosf((((float) M_PI) * (2.0f * uy)));
	} else {
		tmp = cosf(asinf((1.0f + (ux * (maxCos - 1.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) <= Float32(0.019999999552965164))
		tmp = Float32(sqrt(Float32(Float32(2.0) * ux)) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	else
		tmp = cos(asin(Float32(Float32(1.0) + Float32(ux * Float32(maxCos - Float32(1.0))))));
	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 ((cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)))) <= single(0.019999999552965164))
		tmp = sqrt((single(2.0) * ux)) * cos((single(pi) * (single(2.0) * uy)));
	else
		tmp = cos(asin((single(1.0) + (ux * (maxCos - single(1.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0199999996

   1. Initial program 39.1%

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

   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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. lift-fma.f32 91.0

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

   9. Applied rewrites 91.0%

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

   10. Taylor expanded in maxCos around 0

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

   12. Applied rewrites 85.8%

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

   if 0.0199999996 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 89.6%

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

   2. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-sqrt.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-pow.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. associate--l+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. cos-asin-rev N/A

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

      13. lower-cos.f32 N/A

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

      14. lower-asin.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

          \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

      20. lift--.f32 78.5

          \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   6. Applied rewrites 78.5%

      \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \cos \sin^{-1} \left(1 + ux \cdot \left(maxCos - 1\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift--.f32 78.6

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

   9. Applied rewrites 78.6%

      \[\leadsto \cos \sin^{-1} \left(1 + ux \cdot \left(maxCos - 1\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995999932289124:\\ \;\;\;\;\cos \sin^{-1} \left(1 + ux \cdot \left(maxCos - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9995999932289124)
     (cos (asin (+ 1.0 (* ux (- maxCos 1.0)))))
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) 1.0))))

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 = cosf(asinf((1.0f + (ux * (maxCos - 1.0f)))));
	} else {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9995999932289124))
		tmp = cos(asin(Float32(Float32(1.0) + Float32(ux * Float32(maxCos - Float32(1.0))))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(1.0));
	end
	return 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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 73.8

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

   4. Applied rewrites 73.8%

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

      1. lift-sqrt.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-pow.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. associate--l+ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. pow2 N/A

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

      12. cos-asin-rev N/A

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

      13. lower-cos.f32 N/A

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

      14. lower-asin.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

          \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

      20. lift--.f32 74.1

          \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   6. Applied rewrites 74.1%

      \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   7. Taylor expanded in ux around 0

      \[\leadsto \cos \sin^{-1} \left(1 + ux \cdot \left(maxCos - 1\right)\right) \]
   8. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift--.f32 74.1

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

   9. Applied rewrites 74.1%

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

   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 37.1%

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

   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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.9

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. lift-fma.f32 92.5

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

   9. Applied rewrites 92.5%

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

   10. Taylor expanded in uy around 0

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

   12. Applied rewrites 75.8%

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

Alternative 13: 49.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sin \cos^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sin (acos (fma maxCos ux (- 1.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(acosf(fmaf(maxCos, ux, (1.0f - ux))));
}

Julia

function code(ux, uy, maxCos)
	return sin(acos(fma(maxCos, ux, Float32(Float32(1.0) - ux))))
end

Derivation

1. Initial program 57.5%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. associate--l+ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. sin-acos-rev N/A

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

   13. lower-sin.f32 N/A

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

   14. lower-acos.f32 N/A

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

   15. lift-*.f32 N/A

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

6. Applied rewrites 49.3%

   \[\leadsto \color{blue}{\sin \cos^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)} \]
7. Add Preprocessing

Alternative 14: 48.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cos (asin (fma maxCos ux (- 1.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(asinf(fmaf(maxCos, ux, (1.0f - ux))));
}

Julia

function code(ux, uy, maxCos)
	return cos(asin(fma(maxCos, ux, Float32(Float32(1.0) - ux))))
end

Derivation

1. Initial program 57.5%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. associate--l+ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. cos-asin-rev N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-asin.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   20. lift--.f32 48.9

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

6. Applied rewrites 48.9%

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

Alternative 15: 47.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \cos \sin^{-1} \left(1 - ux\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (cos (asin (- 1.0 ux))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(asinf((1.0f - 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 = cos(asin((1.0e0 - ux)))
end function

Julia

function code(ux, uy, maxCos)
	return cos(asin(Float32(Float32(1.0) - ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos(asin((single(1.0) - ux)));
end

Derivation

1. Initial program 57.5%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. associate--l+ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. cos-asin-rev N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-asin.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   20. lift--.f32 48.9

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

6. Applied rewrites 48.9%

   \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

7. Taylor expanded in maxCos around 0

   \[\leadsto \cos \sin^{-1} \left(1 - ux\right) \]
8. Step-by-step derivation

   1. lift--.f32 47.3

      \[\leadsto \cos \sin^{-1} \left(1 - ux\right) \]

9. Applied rewrites 47.3%

   \[\leadsto \cos \sin^{-1} \left(1 - ux\right) \]
10. Add Preprocessing

Alternative 16: 19.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \cos \sin^{-1} \left(maxCos \cdot ux\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (cos (asin (* maxCos ux))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(asinf((maxCos * 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 = cos(asin((maxcos * ux)))
end function

Julia

function code(ux, uy, maxCos)
	return cos(asin(Float32(maxCos * ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos(asin((maxCos * ux)));
end

Derivation

1. Initial program 57.5%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. associate--l+ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. cos-asin-rev N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-asin.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   20. lift--.f32 48.9

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

6. Applied rewrites 48.9%

   \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

7. Taylor expanded in maxCos around inf

   \[\leadsto \cos \sin^{-1} \left(maxCos \cdot ux\right) \]
8. Step-by-step derivation

   1. lower-*.f32 19.8

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

9. Applied rewrites 19.8%

   \[\leadsto \cos \sin^{-1} \left(maxCos \cdot ux\right) \]
10. Add Preprocessing

Alternative 17: 6.6% accurate, 6.3× 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 57.5%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

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

Alternative 18: 4.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \cos \sin^{-1} 1 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (cos (asin 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return cosf(asinf(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 = cos(asin(1.0e0))
end function

Julia

function code(ux, uy, maxCos)
	return cos(asin(Float32(1.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos(asin(single(1.0)));
end

Derivation

1. Initial program 57.5%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

   1. lift-sqrt.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. associate--l+ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. pow2 N/A

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

   12. cos-asin-rev N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-asin.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

   20. lift--.f32 48.9

       \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

6. Applied rewrites 48.9%

   \[\leadsto \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right) \]

7. Taylor expanded in ux around 0

   \[\leadsto \cos \sin^{-1} 1 \]
8. Step-by-step derivation

9. Applied rewrites 4.4%

   \[\leadsto \cos \sin^{-1} 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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)))))))