UniformSampleCone, y

Percentage Accurate: 57.7% → 98.3%

Time: 7.3s

Alternatives: 13

Speedup: 4.8×

Specification

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

Math

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

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: 57.7% accurate, 1.0× speedup

Math

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

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.2%

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3%

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

7. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 inf

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

   1. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.2%

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3%

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

7. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-+r+ N/A

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

   5. lower-+.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

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

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

   11. lower-fma.f32 N/A

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

   12. lift-pow.f32 N/A

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

   13. unpow2 N/A

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

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

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

   15. lift--.f32 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-*.f32 N/A

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

   18. lower--.f32 98.2%

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

9. Applied rewrites 98.2%

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

    1. lift-pow.f32 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f32 98.2%

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

11. Applied rewrites 98.2%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(t\_0 - 0\right) \cdot \left(-\left(t\_0 - 2\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (* ux (- 1.0 maxCos))))
  (* (sin (* (* uy 2.0) PI)) (sqrt (* (- t_0 0.0) (- (- t_0 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - maxCos);
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((t_0 - 0.0f) * -(t_0 - 2.0f)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - maxCos))
	return Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(t_0 - Float32(0.0)) * Float32(-Float32(t_0 - Float32(2.0))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = ux * (single(1.0) - maxCos);
	tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt(((t_0 - single(0.0)) * -(t_0 - single(2.0))));
end

Derivation

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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

4. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\color{blue}{ux \cdot \left(1 - maxCos\right)} - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
5. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \color{blue}{\left(1 - maxCos\right)} - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   2. lower--.f32 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(1 - \color{blue}{maxCos}\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

6. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\color{blue}{ux \cdot \left(1 - maxCos\right)} - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

7. Taylor expanded in ux around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right) - 0\right) \cdot \left(-\left(\color{blue}{ux \cdot \left(1 - maxCos\right)} - 2\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right) - 0\right) \cdot \left(-\left(ux \cdot \color{blue}{\left(1 - maxCos\right)} - 2\right)\right)} \]

   2. lower--.f32 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right) - 0\right) \cdot \left(-\left(ux \cdot \left(1 - \color{blue}{maxCos}\right) - 2\right)\right)} \]

9. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right) - 0\right) \cdot \left(-\left(\color{blue}{ux \cdot \left(1 - maxCos\right)} - 2\right)\right)} \]
10. Add Preprocessing

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.2%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   3. lower-*.f32 98.2%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

5. Applied rewrites 98.3%

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

Alternative 5: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

   3. lower-*.f32 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)} \]

5. Applied rewrites 98.3%

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

Alternative 6: 97.1% accurate, 1.1× speedup

Math

\[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(2 - ux\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (sin (* (* uy 2.0) PI))
 (sqrt (* (- (- ux (* maxCos ux)) 0.0) (- 2.0 ux)))))

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((((ux - (maxCos * ux)) - single(0.0)) * (single(2.0) - ux)));
end

Derivation

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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

4. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \color{blue}{\left(2 - ux\right)}} \]
5. Step-by-step derivation

   1. lower--.f32 97.1%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(2 - \color{blue}{ux}\right)} \]

6. Applied rewrites 97.1%

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

Alternative 7: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (sin (* (* uy 2.0) PI))))
  (if (<= maxCos 4.0000000467443897e-7)
    (* t_0 (sqrt (* ux (- 2.0 ux))))
    (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = t_0 * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = sin(((uy * single(2.0)) * single(pi)));
	tmp = single(0.0);
	if (maxCos <= single(4.0000000467443897e-7))
		tmp = t_0 * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = t_0 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   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. Step-by-step derivation

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.3%

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

   6. Applied rewrites 92.3%

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

   if 4.00000005e-7 < maxCos

   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

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.4%

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

   4. Applied rewrites 76.4%

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

Alternative 8: 94.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= maxCos 1.9999999494757503e-5)
  (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))
  (*
   2.0
   (*
    uy
    (*
     PI
     (sqrt (* (- ux (* maxCos ux)) (- (+ 2.0 (* maxCos ux)) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999494757503e-5f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux - (maxCos * ux)) * ((2.0f + (maxCos * ux)) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999494757503e-5))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux - Float32(maxCos * ux)) * Float32(Float32(Float32(2.0) + Float32(maxCos * ux)) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.9999999494757503e-5))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux - (maxCos * ux)) * ((single(2.0) + (maxCos * ux)) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   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. Step-by-step derivation

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.3%

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

   6. Applied rewrites 92.3%

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

   if 1.99999995e-5 < maxCos

   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. Step-by-step derivation

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 81.2%

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

   6. Applied rewrites 81.2%

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

Alternative 9: 81.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.2%

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

6. Applied rewrites 81.2%

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

Alternative 10: 65.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.6%

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 65.7%

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

7. Applied rewrites 65.7%

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

8. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 65.5%

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

10. Applied rewrites 65.5%

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

Alternative 11: 65.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.6%

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 65.7%

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

7. Applied rewrites 65.7%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. sqrt-prod N/A

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

   5. lower-unsound-*.f32 N/A

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

   6. lower-unsound-sqrt.f32 N/A

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

   7. lift--.f32 N/A

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

   8. sub-flip N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

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

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. lower-unsound-sqrt.f32 65.7%

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

9. Applied rewrites 65.7%

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

Alternative 12: 65.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.6%

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 65.7%

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

7. Applied rewrites 65.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 65.7%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 65.7%

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 65.7%

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

9. Applied rewrites 65.7%

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

Alternative 13: 63.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.6%

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 65.7%

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

7. Applied rewrites 65.7%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 63.1%

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

10. Applied rewrites 63.1%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lift-*.f32 N/A

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

    6. lower-*.f32 63.1%

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. count-2-rev N/A

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

    10. lower-+.f32 63.1%

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

    11. lift-*.f32 N/A

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

12. Applied rewrites 63.1%

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

Reproduce

herbie shell --seed 2025212 
(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)))))))