UniformSampleCone, y

Percentage Accurate: 58.0% → 98.3%

Time: 14.9s

Alternatives: 14

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in ux around inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. associate--l+ 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-*r/ 98.4%

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

   7. metadata-eval 98.4%

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

   8. associate-*r/ 98.4%

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

   9. div-sub 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Taylor expanded in ux around 0 98.4%

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

   1. mul-1-neg 98.4%

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

   2. unsub-neg 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in ux around inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. associate--l+ 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-*r/ 98.4%

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

   7. metadata-eval 98.4%

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

   8. associate-*r/ 98.4%

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

   9. div-sub 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in uy around inf 98.3%

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

    1. associate-*l* 98.4%

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

    2. unpow2 98.4%

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

    3. fma-define 98.4%

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

    4. associate-*r/ 98.4%

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

    5. metadata-eval 98.4%

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

    6. unpow2 98.4%

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

    7. sub-neg 98.4%

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

    8. metadata-eval 98.4%

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

11. Simplified 98.4%

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

12. Taylor expanded in maxCos around 0 98.0%

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

    1. associate--l+ 98.0%

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

    2. associate-*r/ 98.0%

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

    3. metadata-eval 98.0%

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

    4. associate-*r/ 98.0%

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

    5. metadata-eval 98.0%

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

14. Simplified 98.0%

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

15. Final simplification 98.0%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 98.1%

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

7. Final simplification 98.1%

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

Alternative 6: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.010499999858438969:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.010499999858438969)
   (*
    (* 2.0 (* uy PI))
    (sqrt
     (* ux (+ 2.0 (- (* maxCos (- (- (* 2.0 ux) (* ux maxCos)) 2.0)) ux)))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.010499999858438969f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((ux * (2.0f + ((maxCos * (((2.0f * ux) - (ux * maxCos)) - 2.0f)) - ux))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.010499999858438969))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(Float32(Float32(Float32(2.0) * ux) - Float32(ux * maxCos)) - Float32(2.0))) - ux)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.010499999858438969))
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((ux * (single(2.0) + ((maxCos * (((single(2.0) * ux) - (ux * maxCos)) - single(2.0))) - ux))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0104999999

   1. Initial program 59.6%

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

   3. Taylor expanded in ux around 0 98.3%

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

      1. associate--l+ 98.4%

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

      2. associate-*r* 98.4%

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

      3. mul-1-neg 98.4%

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

      4. sub-neg 98.4%

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

      5. metadata-eval 98.4%

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

      6. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in maxCos around 0 98.4%

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

   7. Taylor expanded in uy around 0 94.6%

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

   if 0.0104999999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 53.3%

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

      1. associate-*l* 53.3%

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

      2. sub-neg 53.3%

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

      3. +-commutative 53.3%

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

      4. distribute-rgt-neg-in 53.3%

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

      5. fma-define 53.3%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in maxCos around 0 52.1%

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

   6. Taylor expanded in ux around 0 75.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.010499999858438969:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(\left(2 \cdot ux - ux \cdot maxCos\right) - 2\right) - ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in ux around inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. associate--l+ 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-*r/ 98.4%

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

   7. metadata-eval 98.4%

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

   8. associate-*r/ 98.4%

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

   9. div-sub 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in maxCos around 0 93.9%

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

    1. sub-neg 93.9%

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

    2. associate-*r/ 93.9%

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

    3. metadata-eval 93.9%

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

    4. metadata-eval 93.9%

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

11. Simplified 93.9%

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

12. Final simplification 93.9%

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

Alternative 8: 92.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 57.9%

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

3. Simplified 58.0%

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

5. Taylor expanded in maxCos around 0 56.7%

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

   1. neg-mul-1 56.7%

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

   2. sub-neg 56.7%

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

7. Applied egg-rr 56.7%

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

8. Taylor expanded in ux around 0 93.8%

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

   1. neg-mul-1 93.8%

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

   2. unsub-neg 93.8%

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

10. Simplified 93.8%

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

Alternative 9: 81.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Taylor expanded in uy around 0 80.8%

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

8. Final simplification 80.8%

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

Alternative 10: 77.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in ux around inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. associate--l+ 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-*r/ 98.4%

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

   7. metadata-eval 98.4%

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

   8. associate-*r/ 98.4%

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

   9. div-sub 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in maxCos around 0 93.9%

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

    1. sub-neg 93.9%

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

    2. associate-*r/ 93.9%

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

    3. metadata-eval 93.9%

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

    4. metadata-eval 93.9%

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

11. Simplified 93.9%

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

12. Taylor expanded in uy around 0 77.9%

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

13. Final simplification 77.9%

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

Alternative 11: 77.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in ux around inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. associate--l+ 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-*r/ 98.4%

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

   7. metadata-eval 98.4%

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

   8. associate-*r/ 98.4%

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

   9. div-sub 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in maxCos around 0 93.9%

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

    1. sub-neg 93.9%

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

    2. associate-*r/ 93.9%

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

    3. metadata-eval 93.9%

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

    4. metadata-eval 93.9%

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

11. Simplified 93.9%

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

12. Taylor expanded in uy around 0 77.8%

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

13. Final simplification 77.8%

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

Alternative 12: 77.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 98.4%

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

   1. associate--l+ 98.4%

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

   2. associate-*r* 98.4%

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

   3. mul-1-neg 98.4%

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

   4. sub-neg 98.4%

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

   5. metadata-eval 98.4%

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

   6. +-commutative 98.4%

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

5. Simplified 98.4%

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

6. Taylor expanded in ux around inf 98.4%

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

   1. mul-1-neg 98.4%

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

   2. associate--l+ 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. associate-*r/ 98.4%

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

   7. metadata-eval 98.4%

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

   8. associate-*r/ 98.4%

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

   9. div-sub 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

8. Simplified 98.4%

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

9. Taylor expanded in maxCos around 0 93.9%

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

    1. sub-neg 93.9%

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

    2. associate-*r/ 93.9%

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

    3. metadata-eval 93.9%

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

    4. metadata-eval 93.9%

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

11. Simplified 93.9%

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

12. Taylor expanded in uy around 0 77.8%

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

    1. associate-*l* 77.8%

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

    2. sub-neg 77.8%

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

    3. metadata-eval 77.8%

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

    4. +-commutative 77.8%

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

    5. associate-*r/ 77.8%

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

    6. metadata-eval 77.8%

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

14. Simplified 77.8%

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

Alternative 13: 0.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 57.9%

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

3. Simplified 58.0%

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

5. Taylor expanded in uy around 0 51.3%

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

7. Simplified 51.4%

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

8. Taylor expanded in ux around 0 64.4%

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

9. Taylor expanded in maxCos around inf 0.0%

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

    1. *-commutative 0.0%

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

    2. *-commutative 0.0%

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

11. Simplified 0.0%

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

12. Final simplification 0.0%

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

Alternative 14: -0.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 57.9%

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

3. Simplified 58.0%

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

5. Taylor expanded in uy around 0 51.3%

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

7. Simplified 51.4%

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

8. Taylor expanded in maxCos around inf -0.0%

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

   1. associate-*r* -0.0%

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

10. Simplified -0.0%

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

11. Final simplification -0.0%

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

Reproduce

herbie shell --seed 2024157 
(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)))))))