UniformSampleCone, x

Percentage Accurate: 57.6% → 99.0%

Time: 13.7s

Alternatives: 14

Speedup: 2.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r* 99.1%

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

   4. mul-1-neg 99.1%

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

   5. sub-neg 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in maxCos around 0 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r* 99.1%

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

   4. mul-1-neg 99.1%

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

   5. sub-neg 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in maxCos around 0 99.1%

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

7. Taylor expanded in maxCos around 0 98.5%

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

8. Final simplification 98.5%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r* 99.1%

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

   4. mul-1-neg 99.1%

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

   5. sub-neg 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in maxCos around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.6%

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

5. Taylor expanded in maxCos around 0 57.7%

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

6. Taylor expanded in ux around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r* 99.1%

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

   4. mul-1-neg 99.1%

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

   5. sub-neg 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. expm1-log1p-u 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + \left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2}\right) + -2 \cdot maxCos\right)} \]

7. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + \left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2}\right) + -2 \cdot maxCos\right)} \]
8. Step-by-step derivation

   1. expm1-undefine 98.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} - 1\right)} \cdot \sqrt{ux \cdot \left(\left(2 + \left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2}\right) + -2 \cdot maxCos\right)} \]

   2. log1p-undefine 98.8%

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

   3. rem-exp-log 98.8%

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

   4. associate-*r* 98.8%

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

9. Applied egg-rr 98.8%

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

10. Taylor expanded in maxCos around 0 96.9%

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

    1. neg-mul-1 96.9%

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

    2. unsub-neg 96.9%

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

12. Simplified 96.9%

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

13. Final simplification 96.9%

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

Alternative 6: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 5.700000110664405 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right) + maxCos \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 5.700000110664405e-5)
   (* (cos (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (*
     ux
     (+ 2.0 (+ (* ux (+ -1.0 (* maxCos (- 2.0 maxCos)))) (* maxCos -2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 5.700000110664405e-5f) {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (2.0f + ((ux * (-1.0f + (maxCos * (2.0f - maxCos)))) + (maxCos * -2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(5.700000110664405e-5))
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(ux * Float32(Float32(-1.0) + Float32(maxCos * Float32(Float32(2.0) - maxCos)))) + Float32(maxCos * Float32(-2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(5.700000110664405e-5))
		tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * (single(2.0) + ((ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos)))) + (maxCos * single(-2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 5.70000011e-5

   1. Initial program 57.6%

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

   3. Taylor expanded in ux around 0 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. metadata-eval 99.1%

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

      3. associate-*r* 99.1%

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

      4. mul-1-neg 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

   5. Simplified 99.1%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + \left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2}\right) + -2 \cdot maxCos\right)} \]

   7. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + \left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2}\right) + -2 \cdot maxCos\right)} \]

   8. Taylor expanded in maxCos around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*r* 97.9%

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

      3. neg-mul-1 97.9%

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

      4. unsub-neg 97.9%

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

   10. Simplified 97.9%

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

   if 5.70000011e-5 < maxCos

   1. Initial program 60.8%

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

   3. Taylor expanded in ux around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. metadata-eval 99.4%

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

      3. associate-*r* 99.4%

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

      4. mul-1-neg 99.4%

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

      5. sub-neg 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in maxCos around 0 99.4%

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

   7. Taylor expanded in uy around 0 89.5%

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + -1 \cdot \left(ux \cdot \left(1 + maxCos \cdot \left(maxCos - 2\right)\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 5.700000110664405 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right) + maxCos \cdot -2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012500000186264515:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right) + maxCos \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.012500000186264515)
   (sqrt
    (*
     ux
     (+ 2.0 (+ (* ux (+ -1.0 (* maxCos (- 2.0 maxCos)))) (* maxCos -2.0)))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.012500000186264515f) {
		tmp = sqrtf((ux * (2.0f + ((ux * (-1.0f + (maxCos * (2.0f - maxCos)))) + (maxCos * -2.0f)))));
	} else {
		tmp = cosf(((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(0.012500000186264515))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(ux * Float32(Float32(-1.0) + Float32(maxCos * Float32(Float32(2.0) - maxCos)))) + Float32(maxCos * Float32(-2.0))))));
	else
		tmp = Float32(cos(Float32(Float32(uy * 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(0.012500000186264515))
		tmp = sqrt((ux * (single(2.0) + ((ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos)))) + (maxCos * single(-2.0))))));
	else
		tmp = cos(((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)) < 0.0125000002

   1. Initial program 57.8%

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

   3. Taylor expanded in ux around 0 99.5%

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

      1. cancel-sign-sub-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. associate-*r* 99.5%

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

      4. mul-1-neg 99.5%

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

      5. sub-neg 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in maxCos around 0 99.5%

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

   7. Taylor expanded in uy around 0 95.2%

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

   if 0.0125000002 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.8%

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

   3. Taylor expanded in ux around 0 49.1%

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

   4. Taylor expanded in maxCos around 0 76.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012500000186264515:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right) + maxCos \cdot -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 + ((ux * ((-1.0e0) + (maxcos * (2.0e0 - maxcos)))) + (maxcos * (-2.0e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-*r* 99.1%

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

   4. mul-1-neg 99.1%

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

   5. sub-neg 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in maxCos around 0 99.1%

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

7. Taylor expanded in uy around 0 80.3%

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

8. Final simplification 80.3%

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

Alternative 9: 79.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * sqrt((((2.0e0 + (maxcos * (-2.0e0))) / ux) + ((-1.0e0) + (maxcos * (2.0e0 - maxcos)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around inf 98.8%

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

4. Taylor expanded in uy around 0 79.8%

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

   1. associate--r+ 79.8%

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

   2. associate-*r/ 79.8%

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

   3. metadata-eval 79.8%

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

   4. associate-*r/ 79.8%

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

   5. div-sub 79.8%

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

   6. cancel-sign-sub-inv 79.8%

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

   7. metadata-eval 79.8%

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

   8. sub-neg 79.8%

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

   9. metadata-eval 79.8%

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

6. Simplified 79.8%

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

7. Taylor expanded in maxCos around 0 79.8%

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

8. Final simplification 79.8%

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

Alternative 10: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * sqrt((((2.0e0 + (maxcos * (-2.0e0))) / ux) + ((-1.0e0) + (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around inf 98.8%

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

4. Taylor expanded in uy around 0 79.8%

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

   1. associate--r+ 79.8%

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

   2. associate-*r/ 79.8%

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

   3. metadata-eval 79.8%

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

   4. associate-*r/ 79.8%

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

   5. div-sub 79.8%

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

   6. cancel-sign-sub-inv 79.8%

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

   7. metadata-eval 79.8%

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

   8. sub-neg 79.8%

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

   9. metadata-eval 79.8%

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

6. Simplified 79.8%

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

7. Taylor expanded in maxCos around 0 79.5%

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

   1. metadata-eval 79.5%

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

   2. cancel-sign-sub-inv 79.5%

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

9. Simplified 79.5%

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

10. Final simplification 79.5%

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

Alternative 11: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * sqrt(((2.0e0 / ux) + ((-1.0e0) + (maxcos * (2.0e0 - (2.0e0 / ux))))))
end function

Julia

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around inf 98.8%

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

4. Taylor expanded in uy around 0 79.8%

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

   1. associate--r+ 79.8%

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

   2. associate-*r/ 79.8%

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

   3. metadata-eval 79.8%

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

   4. associate-*r/ 79.8%

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

   5. div-sub 79.8%

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

   6. cancel-sign-sub-inv 79.8%

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

   7. metadata-eval 79.8%

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

   8. sub-neg 79.8%

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

   9. metadata-eval 79.8%

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

6. Simplified 79.8%

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

7. Taylor expanded in maxCos around 0 79.5%

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

   1. associate--l+ 79.5%

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

   2. associate-*r/ 79.5%

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

   3. metadata-eval 79.5%

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

   4. associate-*r/ 79.5%

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

   5. metadata-eval 79.5%

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

9. Simplified 79.5%

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

10. Final simplification 79.5%

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

Alternative 12: 75.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.6%

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

5. Taylor expanded in uy around 0 50.3%

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

7. Simplified 50.1%

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

8. Taylor expanded in maxCos around 0 48.2%

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

9. Taylor expanded in ux around 0 74.5%

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

    1. neg-mul-1 74.5%

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

    2. unsub-neg 74.5%

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

11. Simplified 74.5%

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

Alternative 13: 62.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((2.0e0 * ux))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.6%

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

5. Taylor expanded in uy around 0 50.3%

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

7. Simplified 50.1%

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

8. Taylor expanded in maxCos around 0 48.2%

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

9. Taylor expanded in ux around 0 60.4%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
10. Step-by-step derivation

    1. *-commutative 60.4%

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

11. Simplified 60.4%

    \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

12. Final simplification 60.4%

    \[\leadsto \sqrt{2 \cdot ux} \]
13. Add Preprocessing

Alternative 14: 30.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

function code(ux, uy, maxCos)
	return sqrt(ux)
end

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.6%

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

5. Taylor expanded in uy around 0 50.3%

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

7. Simplified 50.1%

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

8. Taylor expanded in maxCos around inf 25.1%

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

   1. *-commutative 25.1%

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

10. Simplified 25.1%

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

11. Taylor expanded in maxCos around 0 30.7%

    \[\leadsto \sqrt{\color{blue}{ux}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))