Beckmann Sample, near normal, slope_y

Percentage Accurate: 56.8% → 98.4%

Time: 4.8s

Alternatives: 9

Speedup: 4.7×

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

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

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: 56.8% accurate, 1.0× speedup

Math

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

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

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* 6.2831854820251465 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((6.2831854820251465f * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(6.2831854820251465) * u2)))
end

Derivation

1. Initial program 56.8%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.003000000026077032:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)} \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* 6.2831854820251465 u2))))
   (if (<= t_0 -0.003000000026077032)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (fma 0.5 (* u1 u1) u1)) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf((6.2831854820251465f * u2));
	float tmp;
	if (t_0 <= -0.003000000026077032f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(fmaf(0.5f, (u1 * u1), u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(6.2831854820251465) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003000000026077032))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(fma(Float32(0.5), Float32(u1 * u1), u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00300000003

   1. Initial program 56.8%

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

   2. Evaluated real constant 56.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   if -0.00300000003 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 56.8%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.9%

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

   4. Applied rewrites 91.9%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) \cdot u1 + 1 \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) \cdot u1\right) \cdot u1 + 1 \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) \cdot \left(u1 \cdot u1\right) + u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. lower-fma.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{3} \cdot u1 + \frac{1}{2}, u1 \cdot u1, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u1, \frac{1}{2}\right), \color{blue}{u1} \cdot u1, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-*.f32 91.9%

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 88.3%

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

   10. Evaluated real constant 88.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.020999999716877937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.020999999716877937)
   (*
    (sqrt (- (log1p (- u1))))
    (* u2 (+ 6.2831854820251465 (* -41.341705691712875 (pow u2 2.0)))))
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.020999999716877937f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * (6.2831854820251465f + (-41.341705691712875f * powf(u2, 2.0f))));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.020999999716877937))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * Float32(Float32(6.2831854820251465) + Float32(Float32(-41.341705691712875) * (u2 ^ Float32(2.0))))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0209999997

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \color{blue}{\left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}\right)}\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + \color{blue}{\frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot \color{blue}{{u2}^{2}}\right)\right) \]

      4. lower-pow.f32 89.3%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{\color{blue}{2}}\right)\right) \]

   7. Applied rewrites 89.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right)} \]

   if 0.0209999997 < u2

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot u1}\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      3. lower-*.f32 88.3%

         \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot \color{blue}{u1}\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.020999999716877937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.020999999716877937)
   (*
    (sqrt (- (log1p (- u1))))
    (* u2 (+ 6.2831854820251465 (* -41.341705691712875 (pow u2 2.0)))))
   (* (sqrt (fma 0.5 (* u1 u1) u1)) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.020999999716877937f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * (6.2831854820251465f + (-41.341705691712875f * powf(u2, 2.0f))));
	} else {
		tmp = sqrtf(fmaf(0.5f, (u1 * u1), u1)) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.020999999716877937))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * Float32(Float32(6.2831854820251465) + Float32(Float32(-41.341705691712875) * (u2 ^ Float32(2.0))))));
	else
		tmp = Float32(sqrt(fma(Float32(0.5), Float32(u1 * u1), u1)) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0209999997

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \color{blue}{\left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}\right)}\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + \color{blue}{\frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot \color{blue}{{u2}^{2}}\right)\right) \]

      4. lower-pow.f32 89.3%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{\color{blue}{2}}\right)\right) \]

   7. Applied rewrites 89.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right)} \]

   if 0.0209999997 < u2

   1. Initial program 56.8%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.9%

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

   4. Applied rewrites 91.9%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) \cdot u1 + 1 \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) \cdot u1\right) \cdot u1 + 1 \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) \cdot \left(u1 \cdot u1\right) + u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. lower-fma.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{3} \cdot u1 + \frac{1}{2}, u1 \cdot u1, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, u1, \frac{1}{2}\right), \color{blue}{u1} \cdot u1, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. lower-*.f32 91.9%

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

   6. Applied rewrites 91.9%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 88.3%

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

   10. Evaluated real constant 88.3%

       \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 94.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.02500000037252903)
   (*
    (sqrt (- (log1p (- u1))))
    (* u2 (+ 6.2831854820251465 (* -41.341705691712875 (pow u2 2.0)))))
   (* (sqrt u1) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.02500000037252903f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * (6.2831854820251465f + (-41.341705691712875f * powf(u2, 2.0f))));
	} else {
		tmp = sqrtf(u1) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.02500000037252903))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * Float32(Float32(6.2831854820251465) + Float32(Float32(-41.341705691712875) * (u2 ^ Float32(2.0))))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0250000004

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \color{blue}{\left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}\right)}\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + \color{blue}{\frac{-762619864465648886625}{18446744073709551616} \cdot {u2}^{2}}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\frac{13176795}{2097152} + \frac{-762619864465648886625}{18446744073709551616} \cdot \color{blue}{{u2}^{2}}\right)\right) \]

      4. lower-pow.f32 89.3%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{\color{blue}{2}}\right)\right) \]

   7. Applied rewrites 89.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(6.2831854820251465 + -41.341705691712875 \cdot {u2}^{2}\right)\right)} \]

   if 0.0250000004 < u2

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 77.2%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.003000000026077032)
   (* (sqrt (- (log1p (- u1)))) (* 6.2831854820251465 u2))
   (* (sqrt u1) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (6.2831854820251465f * u2);
	} else {
		tmp = sqrtf(u1) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(6.2831854820251465) * u2));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00300000003

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\frac{13176795}{2097152} \cdot u2\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 81.6%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(6.2831854820251465 \cdot \color{blue}{u2}\right) \]

   7. Applied rewrites 81.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(6.2831854820251465 \cdot u2\right)} \]

   if 0.00300000003 < u2

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 77.2%

      \[\leadsto \sqrt{\color{blue}{u1}} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.6% accurate, 2.6× speedup

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(6.2831854820251465 \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (* 6.2831854820251465 u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (6.2831854820251465f * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(6.2831854820251465) * u2))
end

Derivation

1. Initial program 56.8%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4%

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

3. Applied rewrites 98.4%

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

4. Evaluated real constant 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\frac{13176795}{2097152} \cdot u2\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 81.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(6.2831854820251465 \cdot \color{blue}{u2}\right) \]

7. Applied rewrites 81.6%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(6.2831854820251465 \cdot u2\right)} \]
8. Add Preprocessing

Alternative 8: 77.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.0002099999983329326:\\ \;\;\;\;6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.0002099999983329326)
     (* 6.2831854820251465 (* u2 (sqrt (- t_0))))
     (* (+ u2 u2) (* (sqrt u1) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.0002099999983329326f) {
		tmp = 6.2831854820251465f * (u2 * sqrtf(-t_0));
	} else {
		tmp = (u2 + u2) * (sqrtf(u1) * ((float) M_PI));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.0002099999983329326))
		tmp = Float32(Float32(6.2831854820251465) * Float32(u2 * sqrt(Float32(-t_0))));
	else
		tmp = Float32(Float32(u2 + u2) * Float32(sqrt(u1) * Float32(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.0002099999983329326))
		tmp = single(6.2831854820251465) * (u2 * sqrt(-t_0));
	else
		tmp = (u2 + u2) * (sqrt(u1) * single(pi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -2.09999998e-4

   1. Initial program 56.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Evaluated real constant 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{13176795}{2097152} \cdot \color{blue}{\left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}}\right) \]

      3. lower-sqrt.f32 N/A

         \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]

      4. lower-neg.f32 N/A

         \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

      5. lower-log.f32 N/A

         \[\leadsto \frac{13176795}{2097152} \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

      6. lower--.f32 50.1%

         \[\leadsto 6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   7. Applied rewrites 50.1%

      \[\leadsto \color{blue}{6.2831854820251465 \cdot \left(u2 \cdot \sqrt{-\log \left(1 - u1\right)}\right)} \]

   if -2.09999998e-4 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 56.8%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.1%

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

         \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 66.8%

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

   7. Applied rewrites 66.8%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f32 66.8%

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

      7. lift-*.f32 N/A

         \[\leadsto \left(u2 + u2\right) \cdot \left(\pi \cdot \sqrt{u1}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

      9. lower-*.f32 66.8%

         \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

   9. Applied rewrites 66.8%

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

Alternative 9: 66.8% accurate, 4.7× speedup

Math

\[\left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.1%

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

4. Applied rewrites 50.1%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

      \[\leadsto 2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{u1}\right)\right) \]

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.8%

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

7. Applied rewrites 66.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 66.8%

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

   7. lift-*.f32 N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\pi \cdot \sqrt{u1}\right) \]

   8. *-commutative N/A

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

   9. lower-*.f32 66.8%

      \[\leadsto \left(u2 + u2\right) \cdot \left(\sqrt{u1} \cdot \pi\right) \]

9. Applied rewrites 66.8%

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

Reproduce

herbie shell --seed 2025191 
(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))))