Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.1%

Time: 12.2s

Alternatives: 15

Speedup: 21.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.0

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

4. Applied rewrites 99.0%

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

Alternative 2: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9991999864578247:\\ \;\;\;\;t\_0 \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9991999864578247)
     (* t_0 (sqrt (- (* u1 (fma u1 -0.5 -1.0)))))
     (*
      (fma -2.0 (* PI (* PI (* u2 u2))) 1.0)
      (sqrt
       (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9991999864578247f) {
		tmp = t_0 * sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f)));
	} else {
		tmp = fmaf(-2.0f, (((float) M_PI) * (((float) M_PI) * (u2 * u2))), 1.0f) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9991999864578247))
		tmp = Float32(t_0 * sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))));
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(pi) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(1.0)) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999199986

   1. Initial program 57.2%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 90.9

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

   5. Applied rewrites 90.9%

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

   if 0.999199986 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.5%

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 93.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9991999864578247:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0012400000123307109:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(-u1 \cdot u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0012400000123307109)
     (sqrt (- (log1p u1) (log1p (- (* u1 u1)))))
     (*
      (cos t_0)
      (sqrt
       (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0012400000123307109f) {
		tmp = sqrtf((log1pf(u1) - log1pf(-(u1 * u1))));
	} else {
		tmp = cosf(t_0) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0012400000123307109))
		tmp = sqrt(Float32(log1p(u1) - log1p(Float32(-Float32(u1 * u1)))));
	else
		tmp = Float32(cos(t_0) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00124000001

   1. Initial program 57.7%

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-log1p.f32 N/A

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

      4. lower-log1p.f32 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(\color{blue}{\mathsf{neg}\left({u1}^{2}\right)}\right)} \]

      6. unpow2 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-*.f32 N/A

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

      9. lower-neg.f32 99.0

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

   7. Applied rewrites 99.0%

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

   if 0.00124000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 95.3

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

   5. Applied rewrites 95.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0012400000123307109:\\ \;\;\;\;\sqrt{\mathsf{log1p}\left(u1\right) - \mathsf{log1p}\left(-u1 \cdot u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.03999999910593033)
     (*
      (fma -2.0 (* PI (* PI (* u2 u2))) 1.0)
      (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))
     (* (cos t_0) (sqrt (fma u1 (* u1 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.03999999910593033f) {
		tmp = fmaf(-2.0f, (((float) M_PI) * (((float) M_PI) * (u2 * u2))), 1.0f) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	} else {
		tmp = cosf(t_0) * sqrtf(fmaf(u1, (u1 * 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.03999999910593033))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(pi) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(1.0)) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	else
		tmp = Float32(cos(t_0) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0399999991

   1. Initial program 57.5%

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 93.8%

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

   if 0.0399999991 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 90.9

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

   5. Applied rewrites 90.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.03999999910593033:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 94.2

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

5. Applied rewrites 94.2%

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

6. Final simplification 94.2%

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

Alternative 6: 92.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 92.4

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

7. Applied rewrites 92.4%

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

8. Final simplification 92.4%

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

Alternative 7: 92.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 92.4

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

5. Applied rewrites 92.4%

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

6. Final simplification 92.4%

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

Alternative 8: 90.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.07999999821186066:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.07999999821186066)
     (*
      (fma -2.0 (* PI (* PI (* u2 u2))) 1.0)
      (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.07999999821186066f) {
		tmp = fmaf(-2.0f, (((float) M_PI) * (((float) M_PI) * (u2 * u2))), 1.0f) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.07999999821186066))
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(pi) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(1.0)) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	else
		tmp = Float32(cos(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0799999982

   1. Initial program 57.7%

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 93.8%

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

   if 0.0799999982 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.7%

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

   4. Applied rewrites 76.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 78.3

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

   7. Applied rewrites 78.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07999999821186066:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

7. Applied rewrites 89.0%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 84.8%

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

11. Final simplification 84.8%

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

Alternative 10: 82.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

7. Applied rewrites 89.0%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 83.2%

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

11. Final simplification 83.2%

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

Alternative 11: 76.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.004999999888241291)
   (fma (sqrt (* u1 (* u1 u1))) 0.25 (sqrt u1))
   (* (fma -2.0 (* PI (* PI (* u2 u2))) 1.0) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.004999999888241291f) {
		tmp = fmaf(sqrtf((u1 * (u1 * u1))), 0.25f, sqrtf(u1));
	} else {
		tmp = fmaf(-2.0f, (((float) M_PI) * (((float) M_PI) * (u2 * u2))), 1.0f) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.004999999888241291))
		tmp = fma(sqrt(Float32(u1 * Float32(u1 * u1))), Float32(0.25), sqrt(u1));
	else
		tmp = Float32(fma(Float32(-2.0), Float32(Float32(pi) * Float32(Float32(pi) * Float32(u2 * u2))), Float32(1.0)) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00499999989

   1. Initial program 57.4%

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in u1 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cos.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-sqrt.f32 87.7

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} + \color{blue}{\frac{1}{4} \cdot \sqrt{{u1}^{3}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \color{blue}{0.25}, \sqrt{u1}\right) \]

   if 0.00499999989 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.6%

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 57.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \pi \cdot \left(\pi \cdot \left(u2 \cdot u2\right)\right), 1\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004999999888241291:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.004999999888241291)
   (fma (sqrt (* u1 (* u1 u1))) 0.25 (sqrt u1))
   (* (sqrt u1) (fma (* PI PI) (* -2.0 (* u2 u2)) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.004999999888241291f) {
		tmp = fmaf(sqrtf((u1 * (u1 * u1))), 0.25f, sqrtf(u1));
	} else {
		tmp = sqrtf(u1) * fmaf((((float) M_PI) * ((float) M_PI)), (-2.0f * (u2 * u2)), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.004999999888241291))
		tmp = fma(sqrt(Float32(u1 * Float32(u1 * u1))), Float32(0.25), sqrt(u1));
	else
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00499999989

   1. Initial program 57.4%

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in u1 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

      4. lower-cos.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-sqrt.f32 87.7

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} + \color{blue}{\frac{1}{4} \cdot \sqrt{{u1}^{3}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \color{blue}{0.25}, \sqrt{u1}\right) \]

   if 0.00499999989 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.6%

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 57.8%

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

Alternative 13: 79.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

7. Applied rewrites 89.0%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 80.0%

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

11. Final simplification 80.0%

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

Alternative 14: 72.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, 0.25, \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (fma (sqrt (* u1 (* u1 u1))) 0.25 (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf(sqrtf((u1 * (u1 * u1))), 0.25f, sqrtf(u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return fma(sqrt(Float32(u1 * Float32(u1 * u1))), Float32(0.25), sqrt(u1))
end

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 90.4%

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

5. Taylor expanded in u1 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. lower-*.f32 N/A

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

   4. lower-cos.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. cube-mult N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-sqrt.f32 88.7

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

7. Applied rewrites 88.7%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} + \color{blue}{\frac{1}{4} \cdot \sqrt{{u1}^{3}}} \]
9. Step-by-step derivation

10. Applied rewrites 73.4%

    \[\leadsto \mathsf{fma}\left(\sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \color{blue}{0.25}, \sqrt{u1}\right) \]
11. Add Preprocessing

Alternative 15: 64.4% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

4. Applied rewrites 74.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]

   2. lower-log1p.f32 64.3

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

7. Applied rewrites 64.3%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
9. Step-by-step derivation

10. Applied rewrites 65.8%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))