Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.3%

Time: 13.5s

Alternatives: 19

Speedup: 8.9×

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: 57.7% 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: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.8%

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.8%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9700000286102295)
   (*
    (* (fma (* (* u2 u2) -1.3333333333333333) (* (* PI PI) PI) (* PI 2.0)) u2)
    (sqrt (- (log1p (- u1)))))
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* PI 2.0) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9700000286102295f) {
		tmp = (fmaf(((u2 * u2) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * u2) * sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) * 2.0f) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9700000286102295))
		tmp = Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * u2) * sqrt(Float32(-log1p(Float32(-u1)))));
	else
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.970000029

   1. Initial program 97.8%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. unpow3 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. lower-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right), 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. lower-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right) \cdot 2}\right) \cdot u2\right) \]

      15. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right) \cdot 2}\right) \cdot u2\right) \]

      16. lower-PI.f32 96.8

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \pi, \color{blue}{\pi} \cdot 2\right) \cdot u2\right) \]

   7. Applied rewrites 96.8%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot u2\right)} \]

   if 0.970000029 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 51.8%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9649999737739563:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9649999737739563)
   (*
    (*
     (fma (* (* u2 u2) -1.3333333333333333) (* (* PI PI) PI) (* PI 2.0))
     (sqrt (- (log1p (- u1)))))
    u2)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* PI 2.0) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9649999737739563f) {
		tmp = (fmaf(((u2 * u2) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * sqrtf(-log1pf(-u1))) * u2;
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) * 2.0f) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9649999737739563))
		tmp = Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * sqrt(Float32(-log1p(Float32(-u1))))) * u2);
	else
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.964999974

   1. Initial program 98.0%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 96.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\log \left(1 - u1\right)}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. neg-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. lower-log.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      5. lower-/.f32 95.1

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   7. Applied rewrites 95.1%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right) \cdot u2} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right) \cdot u2} \]

   10. Applied rewrites 96.8%

       \[\leadsto \color{blue}{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right)\right) \cdot u2} \]

   if 0.964999974 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9649999737739563:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\pi \cdot \pi, \left(u2 \cdot u2\right) \cdot -1.3333333333333333, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9700000286102295)
   (*
    (* (* (fma (* PI PI) (* (* u2 u2) -1.3333333333333333) 2.0) PI) u2)
    (sqrt (- (log1p (- u1)))))
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (* PI 2.0) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9700000286102295f) {
		tmp = ((fmaf((((float) M_PI) * ((float) M_PI)), ((u2 * u2) * -1.3333333333333333f), 2.0f) * ((float) M_PI)) * u2) * sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(((((float) M_PI) * 2.0f) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9700000286102295))
		tmp = Float32(Float32(Float32(fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(2.0)) * Float32(pi)) * u2) * sqrt(Float32(-log1p(Float32(-u1)))));
	else
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.970000029

   1. Initial program 97.8%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 96.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]
   6. Step-by-step derivation

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\log \left(1 - u1\right)}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 - u1\right)}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. lift-neg.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      5. lower-log1p.f32 96.7

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   7. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(\mathsf{fma}\left(\pi \cdot \pi, -1.3333333333333333 \cdot \left(u2 \cdot u2\right), 2\right) \cdot \pi\right) \cdot u2\right)} \]

   if 0.970000029 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 51.8%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 98.5

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9700000286102295:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\pi \cdot \pi, \left(u2 \cdot u2\right) \cdot -1.3333333333333333, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.00039999998989515007:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.1599999964237213:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.00039999998989515007)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (if (<= t_0 0.1599999964237213)
       (*
        (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2)
        (sqrt
         (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1)))
       (* (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 (t_0 <= 0.00039999998989515007f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} else if (t_0 <= 0.1599999964237213f) {
		tmp = ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2) * sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1));
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00039999998989515007))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	elseif (t_0 <= Float32(0.1599999964237213))
		tmp = Float32(Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2) * sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)));
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.9999999e-4

   1. Initial program 61.0%

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 91.7

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

   7. Applied rewrites 91.7%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-neg.f32 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{0 - \left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. lift-+.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{0 - \color{blue}{\left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. associate--r+ N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{\left(0 - -1\right) - u1 \cdot u1}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. flip-- N/A

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

      13. sub-neg N/A

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

      14. lift-neg.f32 N/A

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

      15. lower-log1p.f32 98.7

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

   9. Applied rewrites 98.7%

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

   if 3.9999999e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.159999996

   1. Initial program 61.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 61.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 91.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   8. Applied rewrites 91.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   if 0.159999996 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 45.4%

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 84.0

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

   7. Applied rewrites 84.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.00039999998989515007:\\ \;\;\;\;\left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{elif}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.1599999964237213:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.8999999761581421:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= (- 1.0 u1) 0.8999999761581421)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) 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 ((1.0f - u1) <= 0.8999999761581421f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.8999999761581421))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	else
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.899999976

   1. Initial program 98.6%

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 88.1

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

   7. Applied rewrites 88.1%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-neg.f32 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{0 - \left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. lift-+.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{0 - \color{blue}{\left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. associate--r+ N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{\left(0 - -1\right) - u1 \cdot u1}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. flip-- N/A

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

      13. sub-neg N/A

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

      14. lift-neg.f32 N/A

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

      15. lower-log1p.f32 88.3

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

   9. Applied rewrites 88.3%

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

   if 0.899999976 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 53.6%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.8999999761581421:\\ \;\;\;\;\left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.002199999988079071:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.002199999988079071)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (* (sqrt (* (fma 0.5 u1 1.0) 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 (t_0 <= 0.002199999988079071f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002199999988079071))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * 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) < 0.0022

   1. Initial program 60.0%

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 91.5

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

   7. Applied rewrites 91.5%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-neg.f32 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{0 - \left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. lift-+.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{0 - \color{blue}{\left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. associate--r+ N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{\left(0 - -1\right) - u1 \cdot u1}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. flip-- N/A

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

      13. sub-neg N/A

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

      14. lift-neg.f32 N/A

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

      15. lower-log1p.f32 98.3

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

   9. Applied rewrites 98.3%

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

   if 0.0022 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.4

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

   5. Applied rewrites 88.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.002199999988079071:\\ \;\;\;\;\left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9819999933242798:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= (- 1.0 u1) 0.9819999933242798)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (*
      (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) 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 ((1.0f - u1) <= 0.9819999933242798f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9819999933242798))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	else
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.981999993

   1. Initial program 97.1%

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 87.5

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

   7. Applied rewrites 87.5%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. lift-log1p.f32 N/A

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

      4. diff-log N/A

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

      5. lift-neg.f32 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{0 - \left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. lift-+.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{0 - \color{blue}{\left(-1 + u1 \cdot u1\right)}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. associate--r+ N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(\frac{\color{blue}{\left(0 - -1\right) - u1 \cdot u1}}{1 + u1}\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f32 N/A

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

      12. flip-- N/A

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

      13. sub-neg N/A

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

      14. lift-neg.f32 N/A

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

      15. lower-log1p.f32 88.4

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

   9. Applied rewrites 88.4%

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

   if 0.981999993 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 50.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9819999933242798:\\ \;\;\;\;\left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.1599999964237213:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.1599999964237213)
     (*
      (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2)
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1)))
     (* (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 (t_0 <= 0.1599999964237213f) {
		tmp = ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2) * sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1));
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.1599999964237213))
		tmp = Float32(Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2) * sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)));
	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) < 0.159999996

   1. Initial program 61.3%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 61.0%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      10. lower-fma.f32 92.0

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   8. Applied rewrites 92.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   if 0.159999996 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 45.4%

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 84.0

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

   7. Applied rewrites 84.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.1599999964237213:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.002199999988079071)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      t_0)
     (*
      (sqrt (* (fma 0.5 u1 1.0) u1))
      (*
       (fma (* (* u2 u2) -1.3333333333333333) (* (* PI PI) PI) (* PI 2.0))
       u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) * 2.0f) * u2;
	float tmp;
	if (t_0 <= 0.002199999988079071f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * (fmaf(((u2 * u2) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0022

   1. Initial program 60.0%

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 91.5

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      10. lower-fma.f32 92.3

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

   10. Applied rewrites 92.3%

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

   if 0.0022 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.9%

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 53.3

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

   7. Applied rewrites 53.3%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 51.7

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

   10. Applied rewrites 51.7%

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

   11. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

       2. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

       3. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\color{blue}{\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

       5. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       6. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       7. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       8. unpow3 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       9. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       10. lower-*.f32 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right), 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       11. lower-PI.f32 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       12. lower-PI.f32 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right), 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       13. lower-PI.f32 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

       14. *-commutative N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right) \cdot 2}\right) \cdot u2\right) \]

       15. lower-*.f32 N/A

           \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right) \cdot 2}\right) \cdot u2\right) \]

       16. lower-PI.f32 66.9

           \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \pi, \color{blue}{\pi} \cdot 2\right) \cdot u2\right) \]

   13. Applied rewrites 66.9%

       \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot u2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.002199999988079071)
     (*
      (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
      t_0)
     (*
      (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2)
      (sqrt (* (fma 0.5 u1 1.0) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) * 2.0f) * u2;
	float tmp;
	if (t_0 <= 0.002199999988079071f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2) * sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2) * sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0022

   1. Initial program 60.0%

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 91.5

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      10. lower-fma.f32 92.3

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

   10. Applied rewrites 92.3%

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

   if 0.0022 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{1}{2} \cdot u1\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + \frac{1}{2} \cdot u1\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(\frac{1}{2} \cdot u1 + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. lower-fma.f32 66.8

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right)} \cdot u1} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   8. Applied rewrites 66.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2)
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2) * sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2) * sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)))
end

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

5. Applied rewrites 55.8%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   6. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   10. lower-fma.f32 84.6

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

8. Applied rewrites 84.6%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

9. Final simplification 84.6%

   \[\leadsto \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 14: 83.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2)
  (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2) * sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2) * sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)))
end

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

   4. associate-*r* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

5. Applied rewrites 55.8%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]

6. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{3} \cdot u1, u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   6. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot u1 + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

   7. lower-fma.f32 82.9

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

8. Applied rewrites 82.9%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

9. Final simplification 82.9%

   \[\leadsto \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 15: 80.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= t_0 0.02500000037252903)
     (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)) t_0)
     (*
      (sqrt u1)
      (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) * 2.0f) * u2;
	float tmp;
	if (t_0 <= 0.02500000037252903f) {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.02500000037252903))
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0250000004

   1. Initial program 60.7%

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 88.5

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

   10. Applied rewrites 88.5%

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

   if 0.0250000004 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 42.0%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\log \left(1 - u1\right)}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. neg-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. lower-log.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      5. lower-/.f32 39.9

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   7. Applied rewrites 39.9%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f32 53.8

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   10. Applied rewrites 53.8%

       \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\pi \cdot 2\right) \cdot u2 \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9999985098838806:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9999985098838806)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (* (* PI 2.0) u2))
   (*
    (sqrt u1)
    (* (* (fma (* (* u2 u2) -1.3333333333333333) (* PI PI) 2.0) PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9999985098838806f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf(u1) * ((fmaf(((u2 * u2) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9999985098838806))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(fma(Float32(Float32(u2 * u2) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * u2));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.99999851

   1. Initial program 81.5%

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.5

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

   7. Applied rewrites 76.5%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      10. lower-fma.f32 76.7

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

   10. Applied rewrites 76.7%

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

   if 0.99999851 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 25.7%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot u2\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right) \cdot u2\right) \]

      4. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right) \cdot u2\right)} \]

   5. Applied rewrites 25.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      2. lift-log.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\log \left(1 - u1\right)}\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      3. neg-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      4. lower-log.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]

      5. lower-/.f32 22.2

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   7. Applied rewrites 22.2%

      \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1}{1 - u1}\right)}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-4}{3}, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 2\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f32 86.5

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]

   10. Applied rewrites 86.5%

       \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(\left(\pi \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9999985098838806:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.8%

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 77.1

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

7. Applied rewrites 77.1%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 76.1

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

10. Applied rewrites 76.1%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \]
11. Add Preprocessing

Alternative 18: 74.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.8%

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

4. Applied rewrites 91.6%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\left(\log \left(\mathsf{neg}\left(\left(-1 + u1 \cdot u1\right)\right)\right) - \mathsf{log1p}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 77.1

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

7. Applied rewrites 77.1%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 73.3

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

10. Applied rewrites 73.3%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \]
11. Add Preprocessing

Alternative 19: 66.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

4. Applied rewrites 74.0%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 63.6

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

7. Applied rewrites 63.6%

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

8. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 65.2

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

10. Applied rewrites 65.2%

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

11. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024235 
(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))))