Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.2% → 98.4%

Time: 16.3s

Alternatives: 10

Speedup: 2.9×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \pi}\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(t\_0 \cdot \left(u2 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (* 2.0 PI))))
   (* (sqrt (- (log1p (- u1)))) (sin (* t_0 (* u2 t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((2.0f * ((float) M_PI)));
	return sqrtf(-log1pf(-u1)) * sinf((t_0 * (u2 * t_0)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(Float32(2.0) * Float32(pi)))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(t_0 * Float32(u2 * t_0))))
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 62.2%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-exp-log 95.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(e^{\log \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)} \]

   2. *-commutative 95.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(e^{\log \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)}}\right) \]

   3. associate-*r* 95.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(e^{\log \color{blue}{\left(\left(u2 \cdot 2\right) \cdot \pi\right)}}\right) \]

6. Applied egg-rr 95.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(e^{\log \left(\left(u2 \cdot 2\right) \cdot \pi\right)}\right)} \]
7. Step-by-step derivation

   1. rem-exp-log 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(u2 \cdot 2\right) \cdot \pi\right)} \]

   2. associate-*l* 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)} \]

   3. add-sqr-sqrt 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(u2 \cdot \color{blue}{\left(\sqrt{2 \cdot \pi} \cdot \sqrt{2 \cdot \pi}\right)}\right) \]

   4. associate-*r* 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(u2 \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt{2 \cdot \pi}\right)} \]

8. Applied egg-rr 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(u2 \cdot \sqrt{2 \cdot \pi}\right) \cdot \sqrt{2 \cdot \pi}\right)} \]

9. Final simplification 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt{2 \cdot \pi} \cdot \left(u2 \cdot \sqrt{2 \cdot \pi}\right)\right) \]
10. Add Preprocessing

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(u2 \cdot 2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (expm1 (log1p (sin (* PI (* u2 2.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * expm1f(log1pf(sinf((((float) M_PI) * (u2 * 2.0f)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * expm1(log1p(sin(Float32(Float32(pi) * Float32(u2 * Float32(2.0)))))))
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 62.2%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right)} \]

   2. *-commutative 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)}\right)\right) \]

   3. associate-*r* 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\left(u2 \cdot 2\right) \cdot \pi\right)}\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]

7. Final simplification 98.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(u2 \cdot 2\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\sin \left(u2 \cdot \pi\right) \cdot \cos \left(u2 \cdot \pi\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (* 2.0 (* (sin (* u2 PI)) (cos (* u2 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (2.0f * (sinf((u2 * ((float) M_PI))) * cosf((u2 * ((float) M_PI)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(u2 * Float32(pi))) * cos(Float32(u2 * Float32(pi))))))
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 62.2%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   2. sin-2 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

6. Applied egg-rr 98.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]

7. Final simplification 98.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\sin \left(u2 \cdot \pi\right) \cdot \cos \left(u2 \cdot \pi\right)\right)\right) \]
8. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* u2 (* 2.0 PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((u2 * (2.0f * ((float) M_PI))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 62.2%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing

5. Final simplification 98.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \]
6. Add Preprocessing

Alternative 5: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* u2 (* 2.0 PI)))
  (sqrt
   (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 (- 0.3333333333333333 (* u1 -0.25))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f - (u1 * -0.25f))))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) - Float32(u1 * Float32(-0.25))))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(2.0) * single(pi)))) * sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * (single(0.3333333333333333) - (u1 * single(-0.25)))))))));
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 91.8%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 91.8%

   \[\leadsto \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* u2 (* 2.0 PI)))
  (sqrt (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 -0.3333333333333333))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * -0.3333333333333333f))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(-0.3333333333333333))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(2.0) * single(pi)))) * sqrt((u1 * (single(1.0) + (u1 * (single(0.5) - (u1 * single(-0.3333333333333333)))))));
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 89.5%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 89.5%

   \[\leadsto \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)} \]
5. Add Preprocessing

Alternative 7: 87.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 (* 2.0 PI))) (sqrt (* u1 (- 1.0 (* u1 -0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (2.0f * ((float) M_PI)))) * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(2.0) * single(pi)))) * sqrt((u1 * (single(1.0) - (u1 * single(-0.5)))));
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 85.0%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-0.5 \cdot u1 - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

4. Final simplification 85.0%

   \[\leadsto \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 (* 2.0 PI))) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * (2.0f * ((float) M_PI)))) * sqrtf(u1);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * (single(2.0) * single(pi)))) * sqrt(u1);
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 59.7%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}} \]

   2. add-sqr-sqrt 59.7%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   3. sqrt-unprod 59.7%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{\sqrt{\left(-\log \left(1 - u1\right)\right) \cdot \left(-\log \left(1 - u1\right)\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   4. sqr-neg 59.7%

      \[\leadsto e^{\log \left(\sqrt{\sqrt{\color{blue}{\log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   5. sqrt-unprod 1.4%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{\log \left(1 - u1\right)}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   6. add-sqr-sqrt 1.4%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{\log \left(1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   7. sub-neg 1.4%

      \[\leadsto e^{\log \left(\sqrt{\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   8. log1p-undefine -0.0%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   9. add-sqr-sqrt -0.0%

      \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(\color{blue}{\sqrt{-u1} \cdot \sqrt{-u1}}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   10. sqrt-unprod 67.1%

       \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   11. sqr-neg 67.1%

       \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(\sqrt{\color{blue}{u1 \cdot u1}}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   12. sqrt-unprod 67.1%

       \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   13. add-sqr-sqrt 67.1%

       \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(\color{blue}{u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]

   14. *-commutative 67.1%

       \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)}\right)} \]

   15. associate-*r* 67.1%

       \[\leadsto e^{\log \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\left(u2 \cdot 2\right) \cdot \pi\right)}\right)} \]

4. Applied egg-rr 67.1%

   \[\leadsto \color{blue}{e^{\log \left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)}} \]

5. Taylor expanded in u1 around 0 72.7%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 72.7%

      \[\leadsto \sqrt{u1} \cdot \sin \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) \]

   2. associate-*r* 72.7%

      \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)} \]

   3. *-commutative 72.7%

      \[\leadsto \sqrt{u1} \cdot \sin \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)} \]

7. Simplified 72.7%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)} \]

8. Final simplification 72.7%

   \[\leadsto \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1} \]
9. Add Preprocessing

Alternative 9: 66.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(u2 \cdot \pi\right) \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 2.0 (* (* u2 PI) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * ((u2 * ((float) M_PI)) * sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(Float32(u2 * Float32(pi)) * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * ((u2 * single(pi)) * sqrt(u1));
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. associate-*r* -0.0%

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}} \]

   2. unpow2 -0.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

   3. rem-square-sqrt 4.0%

      \[\leadsto \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \color{blue}{-1} \]

   4. *-commutative 4.0%

      \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}\right)} \cdot -1 \]

   5. associate-*l* 4.0%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(\sqrt{u1} \cdot -1\right)} \]

   6. *-commutative 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{u1}\right)} \]

   7. mul-1-neg 4.0%

      \[\leadsto \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \color{blue}{\left(-\sqrt{u1}\right)} \]

5. Simplified 4.0%

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \left(-\sqrt{u1}\right)} \]

6. Taylor expanded in u2 around 0 4.6%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 4.6%

      \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot \pi\right)} \]

8. Simplified 4.6%

   \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot \pi\right)} \]

9. Taylor expanded in u1 around -inf -0.0%

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* -0.0%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

    2. unpow2 -0.0%

       \[\leadsto 2 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)\right) \]

    3. rem-square-sqrt 4.6%

       \[\leadsto 2 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot \color{blue}{-1}\right)\right) \]

11. Simplified 4.6%

    \[\leadsto \color{blue}{2 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot -1\right)\right)} \]
12. Step-by-step derivation

    1. pow1 4.6%

       \[\leadsto 2 \cdot \color{blue}{{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot -1\right)\right)}^{1}} \]

    2. pow1 4.6%

       \[\leadsto 2 \cdot \color{blue}{{\left({\left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot -1\right)\right)}^{1}\right)}^{1}} \]

    3. pow1 4.6%

       \[\leadsto 2 \cdot {\color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\pi \cdot -1\right)\right)}}^{1} \]

    4. associate-*l* 4.6%

       \[\leadsto 2 \cdot {\color{blue}{\left(\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot -1\right)\right)\right)}}^{1} \]

    5. add-sqr-sqrt -0.0%

       \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\sqrt{\pi \cdot -1} \cdot \sqrt{\pi \cdot -1}\right)}\right)\right)}^{1} \]

    6. sqrt-unprod 63.9%

       \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\sqrt{\left(\pi \cdot -1\right) \cdot \left(\pi \cdot -1\right)}}\right)\right)}^{1} \]

    7. *-commutative 63.9%

       \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \sqrt{\color{blue}{\left(-1 \cdot \pi\right)} \cdot \left(\pi \cdot -1\right)}\right)\right)}^{1} \]

    8. *-commutative 63.9%

       \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \sqrt{\left(-1 \cdot \pi\right) \cdot \color{blue}{\left(-1 \cdot \pi\right)}}\right)\right)}^{1} \]

    9. swap-sqr 63.9%

       \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \sqrt{\color{blue}{\left(-1 \cdot -1\right) \cdot \left(\pi \cdot \pi\right)}}\right)\right)}^{1} \]

    10. metadata-eval 63.9%

        \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \sqrt{\color{blue}{1} \cdot \left(\pi \cdot \pi\right)}\right)\right)}^{1} \]

    11. *-un-lft-identity 63.9%

        \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \sqrt{\color{blue}{\pi \cdot \pi}}\right)\right)}^{1} \]

    12. sqrt-unprod 63.9%

        \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}\right)\right)}^{1} \]

    13. add-sqr-sqrt 63.9%

        \[\leadsto 2 \cdot {\left(\sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\pi}\right)\right)}^{1} \]

13. Applied egg-rr 63.9%

    \[\leadsto 2 \cdot \color{blue}{{\left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)}^{1}} \]
14. Step-by-step derivation

    1. unpow1 63.9%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]

15. Simplified 63.9%

    \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \left(u2 \cdot \pi\right)\right)} \]

16. Final simplification 63.9%

    \[\leadsto 2 \cdot \left(\left(u2 \cdot \pi\right) \cdot \sqrt{u1}\right) \]
17. Add Preprocessing

Alternative 10: 7.1% accurate, 310.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

1. Initial program 62.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 62.2%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-define 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right)} \]

   2. *-commutative 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)}\right)\right) \]

   3. associate-*r* 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\left(u2 \cdot 2\right) \cdot \pi\right)}\right)\right) \]

6. Applied egg-rr 98.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right)\right)} \]

8. Applied egg-rr 7.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \sin 0\right)} - 1} \]
9. Step-by-step derivation

   1. log1p-undefine 7.1%

      \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{\mathsf{log1p}\left(u1\right)} \cdot \sin 0\right)}} - 1 \]

   2. rem-exp-log 7.1%

      \[\leadsto \color{blue}{\left(1 + \sqrt{\mathsf{log1p}\left(u1\right)} \cdot \sin 0\right)} - 1 \]

   3. sin-0 7.1%

      \[\leadsto \left(1 + \sqrt{\mathsf{log1p}\left(u1\right)} \cdot \color{blue}{0}\right) - 1 \]

   4. mul0-rgt 7.1%

      \[\leadsto \left(1 + \color{blue}{0}\right) - 1 \]

   5. sin-0 7.1%

      \[\leadsto \left(1 + \color{blue}{\sin 0}\right) - 1 \]

   6. +-commutative 7.1%

      \[\leadsto \color{blue}{\left(\sin 0 + 1\right)} - 1 \]

   7. sin-0 7.1%

      \[\leadsto \left(\color{blue}{0} + 1\right) - 1 \]

   8. metadata-eval 7.1%

      \[\leadsto \color{blue}{1} - 1 \]

   9. 1-exp 7.1%

      \[\leadsto \color{blue}{e^{0}} - 1 \]

   10. metadata-eval 7.1%

       \[\leadsto e^{\color{blue}{{0}^{0.3333333333333333}}} - 1 \]

   11. mul0-rgt -0.0%

       \[\leadsto e^{{\color{blue}{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot 0\right)}}^{0.3333333333333333}} - 1 \]

   12. sin-0 -0.0%

       \[\leadsto e^{{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot \color{blue}{\sin 0}\right)}^{0.3333333333333333}} - 1 \]

   13. rem-cube-cbrt -0.0%

       \[\leadsto e^{{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\sin 0}\right)}^{3}}\right)}^{0.3333333333333333}} - 1 \]

   14. unpow1/3 -0.0%

       \[\leadsto e^{{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot {\color{blue}{\left({\sin 0}^{0.3333333333333333}\right)}}^{3}\right)}^{0.3333333333333333}} - 1 \]

   15. sin-0 -0.0%

       \[\leadsto e^{{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot {\left({\color{blue}{0}}^{0.3333333333333333}\right)}^{3}\right)}^{0.3333333333333333}} - 1 \]

   16. metadata-eval -0.0%

       \[\leadsto e^{{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot {\color{blue}{0}}^{3}\right)}^{0.3333333333333333}} - 1 \]

   17. sin-0 -0.0%

       \[\leadsto e^{{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right)}^{3} \cdot {\color{blue}{\sin 0}}^{3}\right)}^{0.3333333333333333}} - 1 \]

   18. cube-prod -0.0%

       \[\leadsto e^{{\color{blue}{\left({\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \sin 0\right)}^{3}\right)}}^{0.3333333333333333}} - 1 \]

   19. unpow1/3 -0.0%

       \[\leadsto e^{\color{blue}{\sqrt[3]{{\left(\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \sin 0\right)}^{3}}}} - 1 \]

   20. rem-cbrt-cube -0.0%

       \[\leadsto e^{\color{blue}{\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \sin 0}} - 1 \]

10. Simplified 7.1%

    \[\leadsto \color{blue}{0} \]

11. Final simplification 7.1%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))