Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 98.8%

Time: 5.2s

Alternatives: 16

Speedup: 11.4×

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: 57.3% 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: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03200000151991844:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(u2, \pi + \pi, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03200000151991844)
   (*
    (sqrt
     (fma u1 1.0 (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (fma u2 (+ PI PI) (/ PI 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03200000151991844f) {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1)))) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(fmaf(u2, (((float) M_PI) + ((float) M_PI)), (((float) M_PI) / 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03200000151991844))
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(pi) / Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0320000015

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. +-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 94.0%

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

   if 0.0320000015 < u1

   1. Initial program 57.3%

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

      1. lift-cos.f32 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 2: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03200000151991844:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(u2, \pi + \pi, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03200000151991844)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (fma u2 (+ PI PI) (/ PI 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03200000151991844f) {
		tmp = sqrtf((fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), u1, 1.0f) * u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(fmaf(u2, (((float) M_PI) + ((float) M_PI)), (((float) M_PI) / 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03200000151991844))
		tmp = Float32(sqrt(Float32(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(pi) / Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0320000015

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 93.9

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

   4. Applied rewrites 93.9%

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

   if 0.0320000015 < u1

   1. Initial program 57.3%

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

      1. lift-cos.f32 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 3: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.01549999974668026:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right) \cdot u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(u2, \pi + \pi, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.01549999974668026)
   (*
    (sqrt (fma u1 1.0 (* u1 (* (fma 0.3333333333333333 u1 0.5) u1))))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (fma u2 (+ PI PI) (/ PI 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.01549999974668026f) {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(0.3333333333333333f, u1, 0.5f) * u1)))) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(fmaf(u2, (((float) M_PI) + ((float) M_PI)), (((float) M_PI) / 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.01549999974668026))
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(Float32(0.3333333333333333), u1, Float32(0.5)) * u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(pi) / Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0154999997

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. +-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in u1 around 0

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

   9. Applied rewrites 92.2%

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

   if 0.0154999997 < u1

   1. Initial program 57.3%

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

      1. lift-cos.f32 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 4: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.01549999974668026:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(u2, \pi + \pi, \frac{\pi}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.01549999974668026)
   (*
    (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (sin (fma u2 (+ PI PI) (/ PI 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.01549999974668026f) {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf(fmaf(u2, (((float) M_PI) + ((float) M_PI)), (((float) M_PI) / 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.01549999974668026))
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(fma(u2, Float32(Float32(pi) + Float32(pi)), Float32(Float32(pi) / Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0154999997

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 92.1

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

   4. Applied rewrites 92.1%

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

   if 0.0154999997 < u1

   1. Initial program 57.3%

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

      1. lift-cos.f32 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 5: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.01549999974668026:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.01549999974668026)
   (*
    (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
    (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.01549999974668026f) {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.01549999974668026))
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0154999997

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f32 92.1

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

   4. Applied rewrites 92.1%

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

   if 0.0154999997 < u1

   1. Initial program 57.3%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 6: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0024999999441206455:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0024999999441206455)
   (* (fma (* (sqrt u1) u1) 0.25 (sqrt u1)) (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0024999999441206455f) {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0024999999441206455))
		tmp = Float32(fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1)) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00249999994

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. pow1/2 N/A

         \[\leadsto \left(\frac{{u1}^{2}}{{u1}^{\frac{1}{2}}} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. pow-div N/A

         \[\leadsto \left({u1}^{\left(2 - \frac{1}{2}\right)} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow2 N/A

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

      8. lower-fma.f32 N/A

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

      9. cube-mult N/A

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

      10. rem-square-sqrt N/A

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

      11. lower-*.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-sqrt.f32 88.6

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

   4. Applied rewrites 88.6%

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

   if 0.00249999994 < u1

   1. Initial program 57.3%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 7: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.0024999999441206455)
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (* 2.0 PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.0024999999441206455f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.0024999999441206455))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00249999994

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.4

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

   4. Applied rewrites 88.4%

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

   if 0.00249999994 < u1

   1. Initial program 57.3%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 57.3

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

   3. Applied rewrites 57.3%

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

Alternative 8: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 0.09799999743700027:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (cos (* (* 2.0 PI) u2))))
   (if (<= (* t_0 t_1) 0.09799999743700027)
     (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_1)
     (*
      t_0
      (fma
       (fma
        (* 0.6666666666666666 (* u2 u2))
        (* (* PI PI) (* PI PI))
        (* (* PI PI) -2.0))
       (* u2 u2)
       1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if ((t_0 * t_1) <= 0.09799999743700027f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_1;
	} else {
		tmp = t_0 * fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_1) <= Float32(0.09799999743700027))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_1);
	else
		tmp = Float32(t_0 * fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0979999974

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f32 88.4

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

   4. Applied rewrites 88.4%

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

   if 0.0979999974 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ t_2 := t\_1 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_2 \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0.125:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (fma
          (fma
           (* 0.6666666666666666 (* u2 u2))
           (* (* PI PI) (* PI PI))
           (* (* PI PI) -2.0))
          (* u2 u2)
          1.0))
        (t_1 (sqrt (- (log (- 1.0 u1)))))
        (t_2 (* t_1 (cos (* (* 2.0 PI) u2)))))
   (if (<= t_2 0.0004199999966658652)
     (* (sqrt u1) (cos (* u2 (+ PI PI))))
     (if (<= t_2 0.125)
       (* (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1)) t_0)
       (* t_1 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	float t_1 = sqrtf(-logf((1.0f - u1)));
	float t_2 = t_1 * cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_2 <= 0.0004199999966658652f) {
		tmp = sqrtf(u1) * cosf((u2 * (((float) M_PI) + ((float) M_PI))));
	} else if (t_2 <= 0.125f) {
		tmp = sqrtf((fmaf(fmaf(0.3333333333333333f, u1, 0.5f), u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0))
	t_1 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_2 = Float32(t_1 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
	tmp = Float32(0.0)
	if (t_2 <= Float32(0.0004199999966658652))
		tmp = Float32(sqrt(u1) * cos(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	elseif (t_2 <= Float32(0.125))
		tmp = Float32(sqrt(Float32(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(t_1 * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 4.19999997e-4

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-cos.f32 76.7

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 76.7

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 76.7

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

   4. Applied rewrites 76.7%

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

   if 4.19999997e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.125

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]

   5. Taylor expanded in u1 around 0

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

      1. neg-log N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 85.1

         \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   7. Applied rewrites 85.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   if 0.125 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ t_2 := t\_1 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_2 \leq 0.0004199999966658652:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (fma
          (fma
           (* 0.6666666666666666 (* u2 u2))
           (* (* PI PI) (* PI PI))
           (* (* PI PI) -2.0))
          (* u2 u2)
          1.0))
        (t_1 (sqrt (- (log (- 1.0 u1)))))
        (t_2 (* t_1 (cos (* (* 2.0 PI) u2)))))
   (if (<= t_2 0.0004199999966658652)
     (* (sqrt u1) (cos (* u2 (+ PI PI))))
     (if (<= t_2 0.05000000074505806)
       (* (sqrt (* (fma 0.5 u1 1.0) u1)) t_0)
       (* t_1 t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	float t_1 = sqrtf(-logf((1.0f - u1)));
	float t_2 = t_1 * cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_2 <= 0.0004199999966658652f) {
		tmp = sqrtf(u1) * cosf((u2 * (((float) M_PI) + ((float) M_PI))));
	} else if (t_2 <= 0.05000000074505806f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0))
	t_1 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_2 = Float32(t_1 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
	tmp = Float32(0.0)
	if (t_2 <= Float32(0.0004199999966658652))
		tmp = Float32(sqrt(u1) * cos(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	elseif (t_2 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * t_0);
	else
		tmp = Float32(t_1 * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 4.19999997e-4

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-cos.f32 76.7

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 76.7

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 76.7

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

   4. Applied rewrites 76.7%

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

   if 4.19999997e-4 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]

   5. Taylor expanded in u1 around 0

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

      1. neg-log N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 81.8

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   7. Applied rewrites 81.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right) \]

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 53.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999549984931946:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.9999549984931946)
   (* (sqrt u1) (cos (* u2 (+ PI PI))))
   (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.9999549984931946f) {
		tmp = sqrtf(u1) * cosf((u2 * (((float) M_PI) + ((float) M_PI))));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.9999549984931946))
		tmp = Float32(sqrt(u1) * cos(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	else
		tmp = sqrt(Float32(-log1p(Float32(-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)) < 0.999954998

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-cos.f32 76.7

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 76.7

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lift-PI.f32 76.7

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

   4. Applied rewrites 76.7%

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

   if 0.999954998 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 49.3

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

   4. Applied rewrites 49.3%

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

      1. lift--.f32 N/A

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

      2. flip3-- N/A

         \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

      3. lower-/.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

      5. lower--.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

      6. unpow3 N/A

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

      7. pow2 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 N/A

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

      11. metadata-eval N/A

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

      12. lower-+.f32 N/A

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

      13. lower-fma.f32 N/A

          \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

      14. lower-*.f32 47.0

          \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   6. Applied rewrites 47.0%

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

      1. lift-log.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

      2. lift-/.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

      3. lift--.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

      4. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

      6. pow3 N/A

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

      7. metadata-eval N/A

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

      8. lift-+.f32 N/A

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

      9. metadata-eval N/A

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

      10. lift-fma.f32 N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

      12. flip3-- N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. fp-cancel-sign-sub-inv N/A

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

      16. mul-1-neg N/A

          \[\leadsto \sqrt{-\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)} \]

      17. lift-neg.f32 N/A

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

      18. lower-log1p.f32 79.6

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

   8. Applied rewrites 79.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := -2 \cdot \left(u2 \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.1850000023841858:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, 1, u1 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1\right)\right)} \cdot \mathsf{fma}\left(t\_1, \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left(\pi \cdot \pi\right) \cdot t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))) (t_1 (* -2.0 (* u2 u2))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.1850000023841858)
     (*
      (sqrt
       (fma
        u1
        1.0
        (* u1 (* (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1))))
      (fma t_1 (* PI PI) 1.0))
     (fma t_1 (* (* PI PI) t_0) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = -2.0f * (u2 * u2);
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.1850000023841858f) {
		tmp = sqrtf(fmaf(u1, 1.0f, (u1 * (fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f) * u1)))) * fmaf(t_1, (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = fmaf(t_1, ((((float) M_PI) * ((float) M_PI)) * t_0), t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = Float32(Float32(-2.0) * Float32(u2 * u2))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.1850000023841858))
		tmp = Float32(sqrt(fma(u1, Float32(1.0), Float32(u1 * Float32(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)) * u1)))) * fma(t_1, Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = fma(t_1, Float32(Float32(Float32(pi) * Float32(pi)) * t_0), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.185000002

   1. Initial program 57.3%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f32 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. +-commutative N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 84.0

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

   9. Applied rewrites 84.0%

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

   if 0.185000002 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

   4. Applied rewrites 52.7%

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

Alternative 13: 79.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-sqrt.f32 49.3

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

4. Applied rewrites 49.3%

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

   1. lift--.f32 N/A

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

   2. flip3-- N/A

      \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

   3. lower-/.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

   5. lower--.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

   6. unpow3 N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f32 N/A

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

   13. lower-fma.f32 N/A

       \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   14. lower-*.f32 47.0

       \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

6. Applied rewrites 47.0%

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

   1. lift-log.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   4. lift-*.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{-\log \left(\frac{1 - \left(u1 \cdot u1\right) \cdot u1}{1 + \mathsf{fma}\left(u1, u1, 1 \cdot u1\right)}\right)} \]

   6. pow3 N/A

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

   7. metadata-eval N/A

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

   8. lift-+.f32 N/A

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

   9. metadata-eval N/A

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

   10. lift-fma.f32 N/A

       \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto \sqrt{-\log \left(\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}\right)} \]

   12. flip3-- N/A

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

   13. *-lft-identity N/A

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

   14. metadata-eval N/A

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

   15. fp-cancel-sign-sub-inv N/A

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

   16. mul-1-neg N/A

       \[\leadsto \sqrt{-\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)} \]

   17. lift-neg.f32 N/A

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

   18. lower-log1p.f32 79.6

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

8. Applied rewrites 79.6%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
9. Add Preprocessing

Alternative 14: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.05000000074505806)
     (sqrt (* (fma 0.5 u1 1.0) u1))
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.05000000074505806f) {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.05000000074505806))
		tmp = sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 49.3

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

   4. Applied rewrites 49.3%

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

   5. Taylor expanded in u1 around 0

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

      1. neg-log N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 72.4

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

   7. Applied rewrites 72.4%

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

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.3%

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

   2. Taylor expanded in u2 around 0

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 49.3

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

   4. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 72.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-sqrt.f32 49.3

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

4. Applied rewrites 49.3%

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

5. Taylor expanded in u1 around 0

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

   1. neg-log N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 72.4

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

7. Applied rewrites 72.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
8. Add Preprocessing

Alternative 16: 64.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\left(-u1\right)} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    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(Float32(-Float32(-u1)))
end

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in u2 around 0

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-sqrt.f32 49.3

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

4. Applied rewrites 49.3%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{--1 \cdot u1} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f32 64.5

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

7. Applied rewrites 64.5%

   \[\leadsto \sqrt{-\left(-u1\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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))))