Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 99.0%

Time: 17.1s

Alternatives: 7

Speedup: 3.1×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.7%

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

   1. sub-neg 62.7%

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

   2. log1p-def 99.2%

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

3. Simplified 99.2%

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

   1. add-exp-log 99.3%

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

   2. associate-*l* 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9997000098228455:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(t\_0 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9997000098228455)
     (* (sqrt u1) (+ 1.0 (+ t_0 -1.0)))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9997000098228455f) {
		tmp = sqrtf(u1) * (1.0f + (t_0 + -1.0f));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9997000098228455))
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(t_0 + Float32(-1.0))));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.99970001

   1. Initial program 61.3%

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

      1. sub-neg 61.3%

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

      2. log1p-def 97.8%

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

   3. Simplified 97.8%

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

      1. add-exp-log 97.9%

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

      2. associate-*l* 97.9%

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

   6. Applied egg-rr 97.9%

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

      1. rem-exp-log 97.8%

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

      2. add-cbrt-cube 97.8%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sqrt[3]{\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}} \]

      3. pow3 97.9%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt[3]{\color{blue}{{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}}} \]

      4. associate-*r* 97.9%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt[3]{{\cos \color{blue}{\left(\left(2 \cdot \pi\right) \cdot u2\right)}}^{3}} \]

      5. *-commutative 97.9%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt[3]{{\cos \color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)}}^{3}} \]

   8. Applied egg-rr 97.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sqrt[3]{{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}^{3}}} \]

   10. Applied egg-rr 70.5%

       \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(1 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) + \sqrt{\mathsf{log1p}\left(u1\right)} \cdot -1} \]
   11. Step-by-step derivation

       1. distribute-lft-out 70.7%

          \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(\left(1 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) + -1\right)} \]

       2. associate-+r+ 70.9%

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right)} \cdot \color{blue}{\left(1 + \left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + -1\right)\right)} \]

       3. +-commutative 70.9%

          \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(1 + \color{blue}{\left(-1 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}\right) \]

   12. Simplified 70.9%

       \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \left(1 + \left(-1 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \]

   13. Taylor expanded in u1 around 0 73.0%

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

   if 0.99970001 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 63.1%

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

      1. sub-neg 63.1%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in u2 around 0 96.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9997000098228455:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9997000098228455:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9997000098228455)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9997000098228455f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9997000098228455))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.99970001

   1. Initial program 61.3%

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

      1. sub-neg 61.3%

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

      2. log1p-def 97.8%

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

   3. Simplified 97.8%

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

      1. neg-mul-1 97.8%

         \[\leadsto \sqrt{\color{blue}{-1 \cdot \mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-udef 61.3%

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

      3. sub-neg 61.3%

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

      4. neg-mul-1 61.3%

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

      5. add-cube-cbrt 61.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)}} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. pow3 61.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right)}^{3}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)}}\right)}^{3}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Taylor expanded in u1 around 0 73.0%

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

   if 0.99970001 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 63.1%

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

      1. sub-neg 63.1%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in u2 around 0 96.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9997000098228455:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 62.7%

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

   1. sub-neg 62.7%

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

   2. log1p-def 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 5: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{\left(-u1\right) \cdot \left(-1 + u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0020000000949949026)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (* (- u1) (+ -1.0 (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0020000000949949026f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((-u1 * (-1.0f + (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0020000000949949026))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(Float32(-u1) * Float32(Float32(-1.0) + Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00200000009

   1. Initial program 62.5%

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

      1. sub-neg 62.5%

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

      2. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in u2 around 0 99.0%

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

   if 0.00200000009 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 62.9%

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

   3. Taylor expanded in u1 around 0 85.8%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative 85.8%

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

      2. *-commutative 85.8%

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

      3. unpow2 85.8%

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

      4. associate-*l* 85.8%

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

      5. distribute-lft-out 85.8%

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

   5. Simplified 85.8%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \left(-1 + u1 \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

Derivation

1. Initial program 62.7%

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

   1. sub-neg 62.7%

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

   2. log1p-def 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in u2 around 0 80.1%

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

6. Final simplification 80.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
7. Add Preprocessing

Alternative 7: 64.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

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

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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 62.7%

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

   1. sub-neg 62.7%

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

   2. log1p-def 99.2%

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

3. Simplified 99.2%

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

   1. add-cbrt-cube 99.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. pow1/3 95.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Applied egg-rr 69.0%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
7. Step-by-step derivation

   1. unpow1/3 70.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Simplified 70.6%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

9. Taylor expanded in u2 around 0 38.0%

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]
10. Step-by-step derivation

    1. log1p-def 60.2%

       \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \]

11. Simplified 60.2%

    \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(u1\right)}} \]

12. Taylor expanded in u1 around 0 62.0%

    \[\leadsto \sqrt{\color{blue}{u1}} \]

13. Final simplification 62.0%

    \[\leadsto \sqrt{u1} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))