UniformSampleCone, y

Percentage Accurate: 57.6% → 98.3%

Time: 10.7s

Alternatives: 21

Speedup: 4.7×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

   1. lift-sin.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. sin-2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-sin.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-cos.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 98.2

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

6. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 98.2

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

10. Applied rewrites 98.2%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-+.f32 98.3

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

7. Applied rewrites 98.3%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. metadata-eval N/A

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

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

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

   6. count-2-rev N/A

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

   7. associate--r+ N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 97.6

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

7. Applied rewrites 97.6%

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

Alternative 5: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.006899999920278788:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(2 - \mathsf{fma}\left(\left(maxCos - 1\right) \cdot ux, maxCos - 1, maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.006899999920278788)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- 2.0 (fma (* (- maxCos 1.0) ux) (- maxCos 1.0) (+ maxCos maxCos)))
      ux)))
   (* (sqrt (* (- 2.0 ux) ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.006899999920278788f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((2.0f - fmaf(((maxCos - 1.0f) * ux), (maxCos - 1.0f), (maxCos + maxCos))) * ux));
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.006899999920278788))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(2.0) - fma(Float32(Float32(maxCos - Float32(1.0)) * ux), Float32(maxCos - Float32(1.0)), Float32(maxCos + maxCos))) * ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00689999992

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right)} \cdot \sqrt{\left(2 - \mathsf{fma}\left(\left(maxCos - 1\right) \cdot ux, maxCos - 1, maxCos + maxCos\right)\right) \cdot ux} \]

   if 0.00689999992 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 91.2

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

   7. Applied rewrites 91.2%

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

Alternative 6: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 96.7

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

10. Applied rewrites 96.7%

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

Alternative 7: 93.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.07999999821186066:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(2 - \mathsf{fma}\left(\left(maxCos - 1\right) \cdot ux, maxCos - 1, maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux + ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.07999999821186066)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- 2.0 (fma (* (- maxCos 1.0) ux) (- maxCos 1.0) (+ maxCos maxCos)))
      ux)))
   (* (sqrt (+ ux ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.07999999821186066f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((2.0f - fmaf(((maxCos - 1.0f) * ux), (maxCos - 1.0f), (maxCos + maxCos))) * ux));
	} else {
		tmp = sqrtf((ux + ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.07999999821186066))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(2.0) - fma(Float32(Float32(maxCos - Float32(1.0)) * ux), Float32(maxCos - Float32(1.0)), Float32(maxCos + maxCos))) * ux)));
	else
		tmp = Float32(sqrt(Float32(ux + ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0799999982

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right)} \cdot \sqrt{\left(2 - \mathsf{fma}\left(\left(maxCos - 1\right) \cdot ux, maxCos - 1, maxCos + maxCos\right)\right) \cdot ux} \]

   if 0.0799999982 < uy

   1. Initial program 56.8%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 72.1

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

   7. Applied rewrites 72.1%

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

Alternative 8: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.003000000026077032:\\ \;\;\;\;\left(\sqrt{ux \cdot \left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right)} \cdot \pi\right) \cdot \left(uy + uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.003000000026077032)
   (*
    (*
     (sqrt
      (*
       ux
       (-
        (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0)
        (+ maxCos maxCos))))
     PI)
    (+ uy uy))
   (* (* (sqrt 2.0) (sin (* (+ uy uy) PI))) (sqrt ux))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.003000000026077032f) {
		tmp = (sqrtf((ux * (fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)))) * ((float) M_PI)) * (uy + uy);
	} else {
		tmp = (sqrtf(2.0f) * sinf(((uy + uy) * ((float) M_PI)))) * sqrtf(ux);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.003000000026077032))
		tmp = Float32(Float32(sqrt(Float32(ux * Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)))) * Float32(pi)) * Float32(uy + uy));
	else
		tmp = Float32(Float32(sqrt(Float32(2.0)) * sin(Float32(Float32(uy + uy) * Float32(pi)))) * sqrt(ux));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00300000003

   1. Initial program 57.8%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 7.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lift--.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f32 95.6

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

   10. Applied rewrites 95.6%

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

   if 0.00300000003 < uy

   1. Initial program 56.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 9: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.003000000026077032:\\ \;\;\;\;\left(\sqrt{ux \cdot \left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right)} \cdot \pi\right) \cdot \left(uy + uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux + ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.003000000026077032)
   (*
    (*
     (sqrt
      (*
       ux
       (-
        (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0)
        (+ maxCos maxCos))))
     PI)
    (+ uy uy))
   (* (sqrt (+ ux ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.003000000026077032f) {
		tmp = (sqrtf((ux * (fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)))) * ((float) M_PI)) * (uy + uy);
	} else {
		tmp = sqrtf((ux + ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.003000000026077032))
		tmp = Float32(Float32(sqrt(Float32(ux * Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)))) * Float32(pi)) * Float32(uy + uy));
	else
		tmp = Float32(sqrt(Float32(ux + ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00300000003

   1. Initial program 57.8%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 57.1%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 7.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lift--.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. count-2-rev N/A

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

      13. lift-+.f32 95.6

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

   10. Applied rewrites 95.6%

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

   if 0.00300000003 < uy

   1. Initial program 56.9%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 10: 81.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. +-commutative N/A

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

   4. pow2 N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. count-2-rev N/A

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

   13. lift-+.f32 81.4

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

10. Applied rewrites 81.4%

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

Alternative 11: 81.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. count-2-rev N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 81.4

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

7. Applied rewrites 81.4%

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

Alternative 12: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos - 1, ux, 1\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_1 \cdot t\_1 \leq 0.9998000264167786:\\ \;\;\;\;\left(\sqrt{1 - t\_0 \cdot t\_0} \cdot \pi\right) \cdot \left(uy + uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos \cdot ux, -2, ux + ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma (- maxCos 1.0) ux 1.0)) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_1 t_1) 0.9998000264167786)
     (* (* (sqrt (- 1.0 (* t_0 t_0))) PI) (+ uy uy))
     (* (sqrt (fma (* maxCos ux) -2.0 (+ ux ux))) (* (+ uy uy) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf((maxCos - 1.0f), ux, 1.0f);
	float t_1 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_1 * t_1) <= 0.9998000264167786f) {
		tmp = (sqrtf((1.0f - (t_0 * t_0))) * ((float) M_PI)) * (uy + uy);
	} else {
		tmp = sqrtf(fmaf((maxCos * ux), -2.0f, (ux + ux))) * ((uy + uy) * ((float) M_PI));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(Float32(maxCos - Float32(1.0)), ux, Float32(1.0))
	t_1 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_1 * t_1) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * Float32(pi)) * Float32(uy + uy));
	else
		tmp = Float32(sqrt(fma(Float32(maxCos * ux), Float32(-2.0), Float32(ux + ux))) * Float32(Float32(uy + uy) * Float32(pi)));
	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))) < 0.999800026

   1. Initial program 88.0%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lift--.f32 75.1

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

   7. Applied rewrites 75.1%

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

   8. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lift--.f32 75.1

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

   10. Applied rewrites 75.1%

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

   if 0.999800026 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 35.4%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 33.0%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. distribute-lft-out N/A

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

      3. count-2-rev N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. count-2-rev N/A

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

      9. associate--r+ N/A

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

      10. lower--.f32 N/A

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

      11. lower--.f32 N/A

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

      12. associate-*r* N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 77.8

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

   7. Applied rewrites 77.8%

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

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift--.f32 N/A

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

      4. associate--l- N/A

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

      5. count-2-rev N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 77.8

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

   9. Applied rewrites 77.8%

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

Alternative 13: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995800256729126:\\ \;\;\;\;\left(\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \pi\right) \cdot \left(uy + uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos \cdot ux, -2, ux + ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9995800256729126)
     (* (* (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))) PI) (+ uy uy))
     (* (sqrt (fma (* maxCos ux) -2.0 (+ ux ux))) (* (+ uy uy) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9995800256729126f) {
		tmp = (sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux)))) * ((float) M_PI)) * (uy + uy);
	} else {
		tmp = sqrtf(fmaf((maxCos * ux), -2.0f, (ux + ux))) * ((uy + uy) * ((float) M_PI));
	}
	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(0.9995800256729126))
		tmp = Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))) * Float32(pi)) * Float32(uy + uy));
	else
		tmp = Float32(sqrt(fma(Float32(maxCos * ux), Float32(-2.0), Float32(ux + ux))) * Float32(Float32(uy + uy) * Float32(pi)));
	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))) < 0.999580026

   1. Initial program 89.6%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lift--.f32 73.0

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. lift--.f32 72.7

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

   10. Applied rewrites 72.7%

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

   if 0.999580026 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 37.6%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. distribute-lft-out N/A

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

      3. count-2-rev N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. count-2-rev N/A

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

      9. associate--r+ N/A

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

      10. lower--.f32 N/A

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

      11. lower--.f32 N/A

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

      12. associate-*r* N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift--.f32 N/A

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

      4. associate--l- N/A

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

      5. count-2-rev N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*r* N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 77.1

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

   9. Applied rewrites 77.1%

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

Alternative 14: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995800256729126:\\ \;\;\;\;\left(\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \pi\right) \cdot \left(uy + uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9995800256729126)
     (* (* (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))) PI) (+ uy uy))
     (* (sqrt (* (fma maxCos -2.0 2.0) ux)) (* (+ uy uy) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9995800256729126f) {
		tmp = (sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux)))) * ((float) M_PI)) * (uy + uy);
	} else {
		tmp = sqrtf((fmaf(maxCos, -2.0f, 2.0f) * ux)) * ((uy + uy) * ((float) M_PI));
	}
	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(0.9995800256729126))
		tmp = Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))) * Float32(pi)) * Float32(uy + uy));
	else
		tmp = Float32(sqrt(Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) * ux)) * Float32(Float32(uy + uy) * Float32(pi)));
	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))) < 0.999580026

   1. Initial program 89.6%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lift--.f32 73.0

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. lift--.f32 72.7

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

   10. Applied rewrites 72.7%

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

   if 0.999580026 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 37.6%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. distribute-lft-out N/A

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

      3. count-2-rev N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. count-2-rev N/A

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

      9. associate--r+ N/A

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

      10. lower--.f32 N/A

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

      11. lower--.f32 N/A

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

      12. associate-*r* N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift--.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate--l- N/A

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

      4. count-2-rev N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 77.1

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

   9. Applied rewrites 77.1%

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

Alternative 15: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \left(uy + uy\right) \cdot \pi\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995800256729126:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* (+ uy uy) PI)))
   (if (<= (* t_0 t_0) 0.9995800256729126)
     (* t_1 (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))
     (* (sqrt (* (fma maxCos -2.0 2.0) ux)) t_1))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = (uy + uy) * ((float) M_PI);
	float tmp;
	if ((t_0 * t_0) <= 0.9995800256729126f) {
		tmp = t_1 * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	} else {
		tmp = sqrtf((fmaf(maxCos, -2.0f, 2.0f) * ux)) * t_1;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = Float32(Float32(uy + uy) * Float32(pi))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9995800256729126))
		tmp = Float32(t_1 * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(sqrt(Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) * ux)) * t_1);
	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))) < 0.999580026

   1. Initial program 89.6%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. count-2-rev N/A

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

      5. lift-+.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. pow2 N/A

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

      10. lift--.f32 N/A

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

      11. lift--.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift--.f32 72.7

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

   7. Applied rewrites 72.7%

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

   if 0.999580026 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 37.6%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. distribute-lft-out N/A

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

      3. count-2-rev N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. count-2-rev N/A

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

      9. associate--r+ N/A

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

      10. lower--.f32 N/A

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

      11. lower--.f32 N/A

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

      12. associate-*r* N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift--.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate--l- N/A

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

      4. count-2-rev N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 77.1

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

   9. Applied rewrites 77.1%

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

Alternative 16: 65.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. count-2-rev N/A

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

   2. distribute-lft-out N/A

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

   3. count-2-rev N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. count-2-rev N/A

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

   9. associate--r+ N/A

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

   10. lower--.f32 N/A

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

   11. lower--.f32 N/A

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

   12. associate-*r* N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. count-2-rev N/A

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

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

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f32 65.9

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

9. Applied rewrites 65.9%

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

Alternative 17: 65.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. count-2-rev N/A

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

   2. distribute-lft-out N/A

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

   3. count-2-rev N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. count-2-rev N/A

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

   9. associate--r+ N/A

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

   10. lower--.f32 N/A

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

   11. lower--.f32 N/A

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

   12. associate-*r* N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

9. Applied rewrites 65.9%

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

Alternative 18: 63.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. count-2-rev N/A

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

   2. distribute-lft-out N/A

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

   3. count-2-rev N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. count-2-rev N/A

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

   9. associate--r+ N/A

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

   10. lower--.f32 N/A

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

   11. lower--.f32 N/A

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

   12. associate-*r* N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 63.6%

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

Alternative 19: 63.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 75.7%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

10. Applied rewrites 72.2%

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

11. Taylor expanded in ux around 0

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

    1. associate-*l* N/A

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

    2. associate-*r* N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. lift-sqrt.f32 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. lift-sqrt.f32 N/A

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

    11. lift-PI.f32 63.4

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

13. Applied rewrites 63.4%

    \[\leadsto \left(\left(\sqrt{ux} \cdot 2\right) \cdot uy\right) \cdot \left(\sqrt{2} \cdot \pi\right) \]
14. Add Preprocessing

Alternative 20: 63.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

   1. count-2-rev N/A

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

   2. distribute-lft-out N/A

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

   3. count-2-rev N/A

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

   4. lower-*.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. count-2-rev N/A

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

   9. associate--r+ N/A

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

   10. lower--.f32 N/A

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

   11. lower--.f32 N/A

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

   12. associate-*r* N/A

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

   13. count-2-rev N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

8. Taylor expanded in maxCos around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f32 63.4

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

10. Applied rewrites 63.4%

    \[\leadsto \sqrt{ux + ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]
11. Add Preprocessing

Alternative 21: 7.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{1 - 1} \cdot \pi\right) \cdot \left(uy + uy\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 50.7%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

   \[\leadsto \left(\sqrt{1 - 1} \cdot \pi\right) \cdot \left(uy + uy\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))