Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.3% → 98.4%

Time: 9.1s

Alternatives: 7

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: 57.3% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

   1. sub-neg 57.8%

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

   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. associate-*l* 98.2%

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

3. Simplified 98.2%

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

Alternative 2: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0005000000237487257)
     (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
     (*
      (sqrt
       (*
        u1
        (+ 1.0 (* u1 (+ 0.5 (* u1 (- 0.3333333333333333 (* u1 -0.25))))))))
      (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0005000000237487257f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f - (u1 * -0.25f)))))))) * sinf(t_0);
	}
	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(0.0005000000237487257))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) - Float32(u1 * Float32(-0.25))))))))) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 5.00000024e-4

   1. Initial program 59.6%

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

      1. sub-neg 59.6%

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

      2. log1p-define 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. add-log-exp 35.2%

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

      2. *-un-lft-identity 35.2%

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

      3. exp-prod 35.2%

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

      4. log-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u2 around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   if 5.00000024e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.6%

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

   3. Taylor expanded in u1 around 0 93.9%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(-0.25 \cdot u1 - 0.3333333333333333\right) - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0005000000237487257)
     (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
     (*
      (sin t_0)
      (sqrt (* u1 (+ 1.0 (* u1 (- 0.5 (* u1 -0.3333333333333333))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0005000000237487257f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f - (u1 * -0.3333333333333333f))))));
	}
	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(0.0005000000237487257))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) - Float32(u1 * Float32(-0.3333333333333333))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 5.00000024e-4

   1. Initial program 59.6%

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

      1. sub-neg 59.6%

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

      2. log1p-define 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. add-log-exp 35.2%

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

      2. *-un-lft-identity 35.2%

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

      3. exp-prod 35.2%

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

      4. log-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u2 around 0 98.6%

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

      1. associate-*r* 98.6%

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

      2. *-commutative 98.6%

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

   9. Simplified 98.6%

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

   if 5.00000024e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.6%

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

   3. Taylor expanded in u1 around 0 92.2%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(u1 \cdot \left(-0.3333333333333333 \cdot u1 - 0.5\right) - 1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.0028200000524520874:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0028200000524520874)
     (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
     (* (sin 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 <= 0.0028200000524520874f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sinf(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(0.0028200000524520874))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(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 #s(literal 2 binary32) (PI.f32)) u2) < 0.0028200001

   1. Initial program 57.8%

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

      1. sub-neg 57.8%

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

      2. log1p-define 98.5%

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

      3. associate-*l* 98.5%

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

   3. Simplified 98.5%

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

      1. add-log-exp 40.7%

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

      2. *-un-lft-identity 40.7%

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

      3. exp-prod 40.7%

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

      4. log-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u2 around 0 97.9%

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

      1. associate-*r* 97.9%

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

      2. *-commutative 97.9%

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

   9. Simplified 97.9%

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

   if 0.0028200001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.0%

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

   3. Taylor expanded in u1 around 0 87.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0028200000524520874:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot u2\right)\\ \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 u2))))
   (if (<= (* u2 (* 2.0 PI)) 0.008999999612569809)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = ((float) M_PI) * (2.0f * u2);
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.008999999612569809f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * u2))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.008999999612569809))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00899999961

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. log1p-define 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. add-log-exp 43.4%

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

      2. *-un-lft-identity 43.4%

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

      3. exp-prod 43.4%

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

      4. log-pow 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in u2 around 0 96.8%

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

      1. associate-*r* 96.8%

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

      2. *-commutative 96.8%

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

   9. Simplified 96.8%

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

   if 0.00899999961 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.6%

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

   3. Taylor expanded in u1 around 0 76.9%

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

      1. mul-1-neg 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in u1 around 0 76.9%

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

      1. associate-*r* 76.9%

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

      2. *-commutative 76.9%

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

   8. Simplified 76.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.008999999612569809:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in u1 around 0 77.0%

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

   1. mul-1-neg 77.0%

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

5. Simplified 77.0%

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

6. Taylor expanded in u1 around 0 77.0%

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

   1. associate-*r* 77.0%

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

   2. *-commutative 77.0%

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

8. Simplified 77.0%

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

Alternative 7: 66.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in u1 around 0 77.0%

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

   1. mul-1-neg 77.0%

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

5. Simplified 77.0%

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

6. Taylor expanded in u2 around 0 66.1%

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

7. Final simplification 66.1%

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

Reproduce

herbie shell --seed 2024096 
(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))))