Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.7% → 99.1%

Time: 15.6s

Alternatives: 6

Speedup: 4.0×

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: 57.7% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.5%

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

   1. sub-neg 58.5%

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

   2. log1p-def 99.3%

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

   3. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 + 2 \cdot \mathsf{fma}\left(\pi, u2, -1\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.00800000037997961)
   (sqrt (- (log1p (- u1))))
   (* (cos (+ 2.0 (* 2.0 (fma PI u2 -1.0)))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.00800000037997961f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((2.0f + (2.0f * fmaf(((float) M_PI), u2, -1.0f)))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.00800000037997961))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) + Float32(Float32(2.0) * fma(Float32(pi), u2, Float32(-1.0))))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00800000038

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. log1p-def 99.6%

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

      3. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in u2 around 0 96.0%

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

   if 0.00800000038 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 60.0%

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

   2. Taylor expanded in u1 around 0 76.9%

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

      1. mul-1-neg 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in u2 around inf 76.9%

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

      1. associate-*r* 76.9%

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

      2. *-commutative 76.9%

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

      3. *-commutative 76.9%

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

   7. Simplified 76.9%

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

      1. associate-*r* 76.9%

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

      2. expm1-log1p-u 76.7%

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

      3. expm1-udef 76.7%

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

      4. log1p-udef 76.9%

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

      5. add-exp-log 76.9%

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

      6. *-commutative 76.9%

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

      7. associate--l+ 76.9%

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

      8. distribute-rgt-in 76.9%

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

      9. metadata-eval 76.9%

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

      10. fma-neg 77.0%

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

      11. metadata-eval 77.0%

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

   9. Applied egg-rr 77.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 + 2 \cdot \mathsf{fma}\left(\pi, u2, -1\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.00800000037997961)
   (sqrt (- (log1p (- u1))))
   (* (sqrt u1) (cos (* PI (* 2.0 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.00800000037997961f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) * cosf((((float) M_PI) * (2.0f * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.00800000037997961))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00800000038

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. log1p-def 99.6%

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

      3. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in u2 around 0 96.0%

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

   if 0.00800000038 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 60.0%

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

   2. Taylor expanded in u1 around 0 76.9%

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

      1. mul-1-neg 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in u2 around inf 76.9%

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

      1. associate-*r* 76.9%

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

      2. *-commutative 76.9%

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

      3. *-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 4: 72.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (fma 0.5 (* u1 u1) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf(0.5f, (u1 * u1), u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(fma(Float32(0.5), Float32(u1 * u1), u1))
end

Derivation

1. Initial program 58.5%

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

   1. flip3-- 55.5%

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

   2. div-inv 55.4%

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

   3. log-prod 55.4%

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

   4. metadata-eval 55.4%

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

   5. pow3 55.4%

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

   6. sub-neg 55.4%

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

   7. distribute-rgt-neg-out 55.4%

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

   8. add-sqr-sqrt -0.0%

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

   9. sqrt-unprod 45.3%

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

   10. sqr-neg 45.3%

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

   11. sqrt-unprod 45.3%

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

   12. add-sqr-sqrt 45.3%

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

   13. log1p-udef 43.7%

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

   14. pow3 43.7%

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left(\color{blue}{{u1}^{3}}\right) + \log \left(\frac{1}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   15. metadata-eval 43.7%

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{\color{blue}{1} + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   16. *-un-lft-identity 43.7%

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{1 + \left(u1 \cdot u1 + \color{blue}{u1}\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   17. fma-def 43.7%

       \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{1 + \color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

3. Applied egg-rr 43.7%

   \[\leadsto \sqrt{-\color{blue}{\left(\mathsf{log1p}\left({u1}^{3}\right) + \log \left(\frac{1}{1 + \mathsf{fma}\left(u1, u1, u1\right)}\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. log-div 43.8%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \color{blue}{\left(\log 1 - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. metadata-eval 43.8%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \left(\color{blue}{0} - \log \left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. log1p-def 86.3%

      \[\leadsto \sqrt{-\left(\mathsf{log1p}\left({u1}^{3}\right) + \left(0 - \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. neg-sub0 86.3%

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

   5. sub-neg 86.3%

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

5. Simplified 86.3%

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

6. Taylor expanded in u2 around 0 42.1%

   \[\leadsto \color{blue}{\sqrt{\log \left({u1}^{2} + \left(1 + u1\right)\right) - \log \left(1 + {u1}^{3}\right)}} \]
7. Step-by-step derivation

   1. log1p-def 40.6%

      \[\leadsto \sqrt{\log \left({u1}^{2} + \left(1 + u1\right)\right) - \color{blue}{\mathsf{log1p}\left({u1}^{3}\right)}} \]

   2. +-commutative 40.6%

      \[\leadsto \sqrt{\log \left({u1}^{2} + \color{blue}{\left(u1 + 1\right)}\right) - \mathsf{log1p}\left({u1}^{3}\right)} \]

   3. associate-+r+ 40.7%

      \[\leadsto \sqrt{\log \color{blue}{\left(\left({u1}^{2} + u1\right) + 1\right)} - \mathsf{log1p}\left({u1}^{3}\right)} \]

   4. unpow2 40.7%

      \[\leadsto \sqrt{\log \left(\left(\color{blue}{u1 \cdot u1} + u1\right) + 1\right) - \mathsf{log1p}\left({u1}^{3}\right)} \]

   5. fma-udef 40.7%

      \[\leadsto \sqrt{\log \left(\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)} + 1\right) - \mathsf{log1p}\left({u1}^{3}\right)} \]

   6. +-commutative 40.7%

      \[\leadsto \sqrt{\log \color{blue}{\left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right)} - \mathsf{log1p}\left({u1}^{3}\right)} \]

   7. log1p-def 73.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right)} - \mathsf{log1p}\left({u1}^{3}\right)} \]

8. Simplified 73.0%

   \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(\mathsf{fma}\left(u1, u1, u1\right)\right) - \mathsf{log1p}\left({u1}^{3}\right)}} \]

9. Taylor expanded in u1 around 0 75.2%

   \[\leadsto \sqrt{\color{blue}{0.5 \cdot {u1}^{2} + u1}} \]
10. Step-by-step derivation

    1. fma-def 75.2%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {u1}^{2}, u1\right)}} \]

    2. unpow2 75.2%

       \[\leadsto \sqrt{\mathsf{fma}\left(0.5, \color{blue}{u1 \cdot u1}, u1\right)} \]

11. Simplified 75.2%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)}} \]

12. Final simplification 75.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)} \]

Alternative 5: 80.0% accurate, 2.0× 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 58.5%

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

   1. sub-neg 58.5%

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

   2. log1p-def 99.3%

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

   3. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u2 around 0 81.8%

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

5. Final simplification 81.8%

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

Alternative 6: 64.8% accurate, 4.0× 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 58.5%

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

2. Taylor expanded in u1 around 0 77.1%

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

   1. mul-1-neg 77.1%

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

4. Simplified 77.1%

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

5. Taylor expanded in u2 around 0 67.0%

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

6. Final simplification 67.0%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023275 
(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))))