UniformSampleCone, y

Percentage Accurate: 57.6% → 98.3%

Time: 13.7s

Alternatives: 13

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: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

   1. distribute-rgt-in 98.3%

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

   2. mul-1-neg 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. *-commutative 98.3%

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

   8. fmm-def 98.3%

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

   9. metadata-eval 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 98.3%

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

7. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* uy (* 2.0 PI)))
  (sqrt
   (-
    (+ (* 2.0 ux) (* -2.0 (* ux maxCos)))
    (* ux (* 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((((2.0f * ux) + (-2.0f * (ux * maxCos))) - (ux * (ux * powf((-1.0f + maxCos), 2.0f)))));
}

Julia

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

   1. distribute-rgt-in 98.3%

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

   2. mul-1-neg 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. *-commutative 98.3%

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

   8. fmm-def 98.3%

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

   9. metadata-eval 98.3%

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in maxCos around 0 98.3%

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

9. Final simplification 98.3%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 98.3%

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

   1. +-commutative 98.3%

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

   2. mul-1-neg 98.3%

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

   3. unsub-neg 98.3%

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

   4. +-commutative 98.3%

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

   5. mul-1-neg 98.3%

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

   6. unsub-neg 98.3%

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

   7. *-commutative 98.3%

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

   8. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 97.4%

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

   1. +-commutative 97.4%

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

   2. mul-1-neg 97.4%

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

   3. unsub-neg 97.4%

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

   4. *-commutative 97.4%

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

8. Simplified 97.4%

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

   1. distribute-rgt-in 97.5%

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

   2. associate-*r* 97.5%

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

   3. *-commutative 97.5%

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

   4. fmm-def 97.5%

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

   5. metadata-eval 97.5%

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

   6. neg-mul-1 97.5%

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

   7. *-commutative 97.5%

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

   8. neg-mul-1 97.5%

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

   9. metadata-eval 97.5%

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

   10. fmm-def 97.5%

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

   11. *-commutative 97.5%

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

   12. mul-1-neg 97.5%

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

   13. fmm-def 97.5%

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

   14. metadata-eval 97.5%

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

10. Applied egg-rr 97.5%

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

    1. metadata-eval 97.5%

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

    2. fmm-def 97.5%

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

    3. distribute-rgt-neg-out 97.5%

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

    4. unsub-neg 97.5%

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

    5. associate-*l* 97.5%

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

    6. fmm-def 97.5%

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

    7. metadata-eval 97.5%

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

12. Simplified 97.5%

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

13. Taylor expanded in uy around inf 97.5%

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

14. Final simplification 97.5%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 97.4%

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

   1. +-commutative 97.4%

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

   2. mul-1-neg 97.4%

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

   3. unsub-neg 97.4%

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

   4. *-commutative 97.4%

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

8. Simplified 97.4%

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

9. Taylor expanded in maxCos around 0 97.5%

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

Alternative 7: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0006000000284984708:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(ux \cdot maxCos\right) - ux\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0006000000284984708)
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+ (* ux (- (* 2.0 (* ux maxCos)) ux)) (* ux (- 2.0 (* 2.0 maxCos)))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0006000000284984708f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf(((ux * ((2.0f * (ux * maxCos)) - ux)) + (ux * (2.0f - (2.0f * maxCos))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0006000000284984708))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(ux * Float32(Float32(Float32(2.0) * Float32(ux * maxCos)) - ux)) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * 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(0.0006000000284984708))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt(((ux * ((single(2.0) * (ux * maxCos)) - ux)) + (ux * (single(2.0) - (single(2.0) * maxCos))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.00000028e-4

   1. Initial program 58.2%

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

      1. associate-*l* 58.2%

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

      2. sub-neg 58.2%

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

      3. +-commutative 58.2%

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

      4. distribute-rgt-neg-in 58.2%

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

      5. fma-define 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in ux around 0 98.4%

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

   6. Taylor expanded in maxCos around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

      4. *-commutative 97.7%

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

   8. Simplified 97.7%

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

      1. distribute-rgt-in 97.8%

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

      2. associate-*r* 97.8%

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

      3. *-commutative 97.8%

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

      4. fmm-def 97.8%

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

      5. metadata-eval 97.8%

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

      6. neg-mul-1 97.8%

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

      7. *-commutative 97.8%

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

      8. neg-mul-1 97.8%

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

      9. metadata-eval 97.8%

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

      10. fmm-def 97.8%

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

      11. *-commutative 97.8%

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

      12. mul-1-neg 97.8%

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

      13. fmm-def 97.8%

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

      14. metadata-eval 97.8%

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

   10. Applied egg-rr 97.8%

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

       1. metadata-eval 97.8%

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

       2. fmm-def 97.8%

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

       3. distribute-rgt-neg-out 97.8%

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

       4. unsub-neg 97.8%

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

       5. associate-*l* 97.8%

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

       6. fmm-def 97.8%

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

       7. metadata-eval 97.8%

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

   12. Simplified 97.8%

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

   13. Taylor expanded in uy around 0 97.5%

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

   if 6.00000028e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 56.3%

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

      1. associate-*l* 56.3%

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

      2. sub-neg 56.3%

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

      3. +-commutative 56.3%

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

      4. distribute-rgt-neg-in 56.3%

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

      5. fma-define 56.4%

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

   3. Simplified 56.5%

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

   5. Taylor expanded in ux around 0 98.1%

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

   6. Taylor expanded in maxCos around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. unsub-neg 91.6%

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

   8. Simplified 91.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0006000000284984708:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(ux \cdot maxCos\right) - ux\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sin (* uy (* 2.0 PI))) (sqrt (* 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 * -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(Float32(2.0) + Float32(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 * single(-2.0))) - ux)));
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 96.6%

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

   1. mul-1-neg 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in ux around 0 96.6%

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

    1. neg-mul-1 96.6%

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

    2. +-commutative 96.6%

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

    3. sub-neg 96.6%

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

    4. neg-mul-1 96.6%

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

    5. fmm-def 96.6%

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

    6. metadata-eval 96.6%

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

    7. fma-undefine 96.6%

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

    8. distribute-neg-in 96.6%

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

    9. distribute-lft-neg-in 96.6%

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

    10. metadata-eval 96.6%

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

    11. metadata-eval 96.6%

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

    12. +-commutative 96.6%

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

11. Simplified 96.6%

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

12. Final simplification 96.6%

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

Alternative 9: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 97.4%

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

   1. +-commutative 97.4%

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

   2. mul-1-neg 97.4%

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

   3. unsub-neg 97.4%

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

   4. *-commutative 97.4%

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

8. Simplified 97.4%

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

   1. distribute-rgt-in 97.5%

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

   2. associate-*r* 97.5%

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

   3. *-commutative 97.5%

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

   4. fmm-def 97.5%

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

   5. metadata-eval 97.5%

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

   6. neg-mul-1 97.5%

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

   7. *-commutative 97.5%

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

   8. neg-mul-1 97.5%

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

   9. metadata-eval 97.5%

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

   10. fmm-def 97.5%

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

   11. *-commutative 97.5%

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

   12. mul-1-neg 97.5%

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

   13. fmm-def 97.5%

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

   14. metadata-eval 97.5%

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

10. Applied egg-rr 97.5%

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

    1. metadata-eval 97.5%

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

    2. fmm-def 97.5%

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

    3. distribute-rgt-neg-out 97.5%

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

    4. unsub-neg 97.5%

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

    5. associate-*l* 97.5%

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

    6. fmm-def 97.5%

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

    7. metadata-eval 97.5%

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

12. Simplified 97.5%

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

13. Taylor expanded in uy around 0 80.2%

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

14. Final simplification 80.2%

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

Alternative 10: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 97.4%

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

   1. +-commutative 97.4%

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

   2. mul-1-neg 97.4%

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

   3. unsub-neg 97.4%

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

   4. *-commutative 97.4%

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

8. Simplified 97.4%

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

9. Taylor expanded in uy around 0 80.2%

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

10. Final simplification 80.2%

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

Alternative 11: 80.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in ux around 0 98.3%

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

6. Taylor expanded in maxCos around 0 96.6%

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

   1. mul-1-neg 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in uy around 0 79.7%

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

    1. associate-*r* 79.7%

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

    2. neg-mul-1 79.7%

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

    3. fmm-def 79.7%

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

    4. metadata-eval 79.7%

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

    5. fma-undefine 79.7%

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

    6. distribute-neg-in 79.7%

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

    7. distribute-lft-neg-in 79.7%

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

    8. metadata-eval 79.7%

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

    9. metadata-eval 79.7%

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

    10. +-commutative 79.7%

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

11. Simplified 79.7%

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

12. Final simplification 79.7%

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

Alternative 12: 66.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in uy around 0 50.1%

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

7. Simplified 50.1%

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

8. Taylor expanded in ux around 0 64.8%

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

   1. *-commutative 64.8%

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

   2. cancel-sign-sub-inv 64.8%

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

   3. metadata-eval 64.8%

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

   4. *-commutative 64.8%

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

10. Simplified 64.8%

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

Alternative 13: 7.1% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. sub-neg 57.5%

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

   3. +-commutative 57.5%

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

   4. distribute-rgt-neg-in 57.5%

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

   5. fma-define 57.6%

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

3. Simplified 57.7%

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

5. Taylor expanded in uy around 0 50.1%

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

7. Simplified 50.1%

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

8. Taylor expanded in ux around 0 7.1%

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

9. Taylor expanded in uy around 0 7.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))))