Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 98.2%

Time: 1.0min

Alternatives: 11

Speedup: 1.3×

• unsoundness detected in the egraph

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), 1\right) \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - 2\right)\right|\right)} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (let* ((t_0 (sin (* 6.2831854820251465 u2))))
  (if (<= u1 0.03500000014901161)
    (*
     (sqrt
      (fma
       (* u1 u1)
       0.5
       (* (fma (* u1 u1) (fma 0.25 u1 0.3333333333333333) 1.0) u1)))
     t_0)
    (*
     (sqrt (- (* 0.5 (log (fabs (+ 1.0 (* u1 (- u1 2.0))))))))
     t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((6.2831854820251465f * u2));
	float tmp;
	if (u1 <= 0.03500000014901161f) {
		tmp = sqrtf(fmaf((u1 * u1), 0.5f, (fmaf((u1 * u1), fmaf(0.25f, u1, 0.3333333333333333f), 1.0f) * u1))) * t_0;
	} else {
		tmp = sqrtf(-(0.5f * logf(fabsf((1.0f + (u1 * (u1 - 2.0f))))))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(6.2831854820251465) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03500000014901161))
		tmp = Float32(sqrt(fma(Float32(u1 * u1), Float32(0.5), Float32(fma(Float32(u1 * u1), fma(Float32(0.25), u1, Float32(0.3333333333333333)), Float32(1.0)) * u1))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-Float32(Float32(0.5) * log(abs(Float32(Float32(1.0) + Float32(u1 * Float32(u1 - Float32(2.0))))))))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0350000001

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.3%

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

   4. Applied rewrites 93.3%

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

   6. Applied rewrites 93.4%

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

   7. Evaluated real constant 93.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), 1\right) \cdot u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   if 0.0350000001 < u1

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

      1. lift-log.f32 N/A

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

      2. *-lft-identity N/A

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

      3. log-prod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\color{blue}{0} + \log \left(\left|1 - u1\right|\right)\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      6. +-lft-identity N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. pow1/2 N/A

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

      9. log-pow N/A

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

      10. lower-*.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower-fabs.f32 N/A

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

      13. sqr-neg-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)}\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      14. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      15. sub-negate-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      16. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(u1 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      17. sub-negate-rev N/A

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

      18. lower-*.f32 N/A

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

      19. lower--.f32 N/A

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

      20. lower--.f32 58.2%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|\color{blue}{1 + u1 \cdot \left(u1 - 2\right)}\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 61.0%

         \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - \color{blue}{2}\right)\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   7. Applied rewrites 61.0%

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

Alternative 2: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - 2\right)\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (if (<= u1 0.03500000014901161)
  (*
   (sqrt
    (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) (* u1 u1) u1))
   (sin (* (* 2.0 PI) u2)))
  (*
   (sqrt (- (* 0.5 (log (fabs (+ 1.0 (* u1 (- u1 2.0))))))))
   (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.03500000014901161f) {
		tmp = sqrtf(fmaf(fmaf(fmaf(0.25f, u1, 0.3333333333333333f), u1, 0.5f), (u1 * u1), u1)) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-(0.5f * logf(fabsf((1.0f + (u1 * (u1 - 2.0f))))))) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03500000014901161))
		tmp = Float32(sqrt(fma(fma(fma(Float32(0.25), u1, Float32(0.3333333333333333)), u1, Float32(0.5)), Float32(u1 * u1), u1)) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-Float32(Float32(0.5) * log(abs(Float32(Float32(1.0) + Float32(u1 * Float32(u1 - Float32(2.0))))))))) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0350000001

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.3%

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

   4. Applied rewrites 93.3%

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

   6. Applied rewrites 93.4%

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

   if 0.0350000001 < u1

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

      1. lift-log.f32 N/A

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

      2. *-lft-identity N/A

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

      3. log-prod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\color{blue}{0} + \log \left(\left|1 - u1\right|\right)\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      6. +-lft-identity N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. pow1/2 N/A

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

      9. log-pow N/A

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

      10. lower-*.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower-fabs.f32 N/A

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

      13. sqr-neg-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)}\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      14. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      15. sub-negate-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      16. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(u1 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      17. sub-negate-rev N/A

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

      18. lower-*.f32 N/A

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

      19. lower--.f32 N/A

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

      20. lower--.f32 58.2%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|\color{blue}{1 + u1 \cdot \left(u1 - 2\right)}\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 61.0%

         \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - \color{blue}{2}\right)\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   7. Applied rewrites 61.0%

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

Alternative 3: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03500000014901161:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - 2\right)\right|\right)} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (let* ((t_0 (sin (* 6.2831854820251465 u2))))
  (if (<= u1 0.03500000014901161)
    (*
     (sqrt
      (*
       u1
       (+
        1.0
        (* u1 (+ 0.5 (* u1 (+ 0.3333333333333333 (* 0.25 u1))))))))
     t_0)
    (*
     (sqrt (- (* 0.5 (log (fabs (+ 1.0 (* u1 (- u1 2.0))))))))
     t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((6.2831854820251465f * u2));
	float tmp;
	if (u1 <= 0.03500000014901161f) {
		tmp = sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f + (0.25f * u1)))))))) * t_0;
	} else {
		tmp = sqrtf(-(0.5f * logf(fabsf((1.0f + (u1 * (u1 - 2.0f))))))) * t_0;
	}
	return tmp;
}

Fortran

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin((6.2831854820251465e0 * u2))
    if (u1 <= 0.03500000014901161e0) then
        tmp = sqrt((u1 * (1.0e0 + (u1 * (0.5e0 + (u1 * (0.3333333333333333e0 + (0.25e0 * u1)))))))) * t_0
    else
        tmp = sqrt(-(0.5e0 * log(abs((1.0e0 + (u1 * (u1 - 2.0e0))))))) * t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(6.2831854820251465) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03500000014901161))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u1)))))))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-Float32(Float32(0.5) * log(abs(Float32(Float32(1.0) + Float32(u1 * Float32(u1 - Float32(2.0))))))))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sin((single(6.2831854820251465) * u2));
	tmp = single(0.0);
	if (u1 <= single(0.03500000014901161))
		tmp = sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * (single(0.3333333333333333) + (single(0.25) * u1)))))))) * t_0;
	else
		tmp = sqrt(-(single(0.5) * log(abs((single(1.0) + (u1 * (u1 - single(2.0)))))))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0350000001

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 93.3%

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

   4. Applied rewrites 93.3%

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

   5. Evaluated real constant 93.3%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]

   if 0.0350000001 < u1

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

      1. lift-log.f32 N/A

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

      2. *-lft-identity N/A

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

      3. log-prod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\color{blue}{0} + \log \left(\left|1 - u1\right|\right)\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      6. +-lft-identity N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. pow1/2 N/A

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

      9. log-pow N/A

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

      10. lower-*.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower-fabs.f32 N/A

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

      13. sqr-neg-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)}\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      14. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      15. sub-negate-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      16. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(u1 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      17. sub-negate-rev N/A

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

      18. lower-*.f32 N/A

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

      19. lower--.f32 N/A

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

      20. lower--.f32 58.2%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|\color{blue}{1 + u1 \cdot \left(u1 - 2\right)}\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 61.0%

         \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - \color{blue}{2}\right)\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   7. Applied rewrites 61.0%

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

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{if}\;\log \left(1 - u1\right) \leq -0.012849999591708183:\\ \;\;\;\;\sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - 2\right)\right|\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (let* ((t_0 (sin (* 6.2831854820251465 u2))))
  (if (<= (log (- 1.0 u1)) -0.012849999591708183)
    (* (sqrt (- (* 0.5 (log (fabs (+ 1.0 (* u1 (- u1 2.0)))))))) t_0)
    (*
     (sqrt (fma (fma 0.3333333333333333 u1 0.5) (* u1 u1) u1))
     t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((6.2831854820251465f * u2));
	float tmp;
	if (logf((1.0f - u1)) <= -0.012849999591708183f) {
		tmp = sqrtf(-(0.5f * logf(fabsf((1.0f + (u1 * (u1 - 2.0f))))))) * t_0;
	} else {
		tmp = sqrtf(fmaf(fmaf(0.3333333333333333f, u1, 0.5f), (u1 * u1), u1)) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(6.2831854820251465) * u2))
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.012849999591708183))
		tmp = Float32(sqrt(Float32(-Float32(Float32(0.5) * log(abs(Float32(Float32(1.0) + Float32(u1 * Float32(u1 - Float32(2.0))))))))) * t_0);
	else
		tmp = Float32(sqrt(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), Float32(u1 * u1), u1)) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0128499996

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

      1. lift-log.f32 N/A

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

      2. *-lft-identity N/A

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

      3. log-prod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

         \[\leadsto \sqrt{-\left(\color{blue}{0} + \log \left(\left|1 - u1\right|\right)\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      6. +-lft-identity N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. pow1/2 N/A

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

      9. log-pow N/A

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

      10. lower-*.f32 N/A

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

      11. lower-log.f32 N/A

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

      12. lower-fabs.f32 N/A

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

      13. sqr-neg-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)}\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      14. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      15. sub-negate-rev N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\color{blue}{\left(u1 - 1\right)} \cdot \left(\mathsf{neg}\left(\left(1 - u1\right)\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      16. lift--.f32 N/A

          \[\leadsto \sqrt{-\frac{1}{2} \cdot \log \left(\left|\left(u1 - 1\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(1 - u1\right)}\right)\right)\right|\right)} \cdot \sin \left(\frac{13176795}{2097152} \cdot u2\right) \]

      17. sub-negate-rev N/A

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

      18. lower-*.f32 N/A

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

      19. lower--.f32 N/A

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

      20. lower--.f32 58.2%

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|\color{blue}{1 + u1 \cdot \left(u1 - 2\right)}\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 61.0%

         \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|1 + u1 \cdot \left(u1 - \color{blue}{2}\right)\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   7. Applied rewrites 61.0%

      \[\leadsto \sqrt{-0.5 \cdot \log \left(\left|\color{blue}{1 + u1 \cdot \left(u1 - 2\right)}\right|\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right) \]

   if -0.0128499996 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.5%

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f32 N/A

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

   6. Applied rewrites 91.6%

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

   7. Evaluated real constant 91.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.01600000075995922:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* 6.2831854820251465 u2))))
  (if (<= t_0 -0.01600000075995922)
    (* (sqrt (- t_0)) t_1)
    (*
     (sqrt (fma (fma 0.3333333333333333 u1 0.5) (* u1 u1) u1))
     t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf((6.2831854820251465f * u2));
	float tmp;
	if (t_0 <= -0.01600000075995922f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf(fmaf(fmaf(0.3333333333333333f, u1, 0.5f), (u1 * u1), u1)) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(6.2831854820251465) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.01600000075995922))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(fma(fma(Float32(0.3333333333333333), u1, Float32(0.5)), Float32(u1 * u1), u1)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0160000008

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

   if -0.0160000008 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 91.5%

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f32 N/A

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

   6. Applied rewrites 91.6%

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

   7. Evaluated real constant 91.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1 \cdot u1, u1\right)} \cdot \sin \left(\color{blue}{6.2831854820251465} \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;u1 \leq 0.00279999990016222:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, 0.5, u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (if (<= u1 0.00279999990016222)
  (* (sqrt (fma (* u1 u1) 0.5 u1)) (sin (* (* 2.0 PI) u2)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00279999990016222f) {
		tmp = sqrtf(fmaf((u1 * u1), 0.5f, u1)) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * sinf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00279999990016222))
		tmp = Float32(sqrt(fma(Float32(u1 * u1), Float32(0.5), u1)) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0027999999

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.7%

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f32 N/A

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

      10. lower-*.f32 87.8%

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

   6. Applied rewrites 87.8%

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

   if 0.0027999999 < u1

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

Alternative 7: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(6.2831854820251465 \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.00279999990016222:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (let* ((t_0 (sin (* 6.2831854820251465 u2))))
  (if (<= u1 0.00279999990016222)
    (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) t_0)
    (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((6.2831854820251465f * u2));
	float tmp;
	if (u1 <= 0.00279999990016222f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Fortran

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = sin((6.2831854820251465e0 * u2))
    if (u1 <= 0.00279999990016222e0) then
        tmp = sqrt((u1 * (1.0e0 + (0.5e0 * u1)))) * t_0
    else
        tmp = sqrt(-log((1.0e0 - u1))) * t_0
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(6.2831854820251465) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00279999990016222))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sin((single(6.2831854820251465) * u2));
	tmp = single(0.0);
	if (u1 <= single(0.00279999990016222))
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * t_0;
	else
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0027999999

   1. Initial program 58.2%

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

   2. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.7%

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

   4. Applied rewrites 87.7%

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

   5. Evaluated real constant 87.7%

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

   if 0.0027999999 < u1

   1. Initial program 58.2%

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

   2. Evaluated real constant 58.2%

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

Alternative 8: 87.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* 6.2831854820251465 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf((6.2831854820251465f * u2));
}

Fortran

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 * (1.0e0 + (0.5e0 * u1)))) * sin((6.2831854820251465e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(6.2831854820251465) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * sin((single(6.2831854820251465) * u2));
end

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 87.7%

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

4. Applied rewrites 87.7%

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

5. Evaluated real constant 87.7%

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

Alternative 9: 76.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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) * sin((6.2831854820251465e0 * u2))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * sin((single(6.2831854820251465) * u2));
end

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.1%

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

4. Applied rewrites 76.1%

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

5. Evaluated real constant 76.1%

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

Alternative 10: 19.1% accurate, 1.4× speedup

Math

\[\sin 6.2831854820251465 \cdot \sqrt{u1} \]

FPCore

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

C

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

Fortran

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 = sin(6.2831854820251465e0) * sqrt(u1)
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin(single(6.2831854820251465)) * sqrt(u1);
end

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.1%

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

4. Applied rewrites 76.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 76.1%

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

6. Applied rewrites 19.1%

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

7. Evaluated real constant 19.1%

   \[\leadsto \sin \color{blue}{6.2831854820251465} \cdot \sqrt{u1} \]
8. Add Preprocessing

Alternative 11: 7.1% accurate, 55.6× speedup

Math

\[0 \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

Fortran

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 = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.1%

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

4. Applied rewrites 76.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 76.1%

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

6. Applied rewrites 19.1%

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

7. Evaluated real constant 7.1%

   \[\leadsto \color{blue}{0} \cdot \sqrt{u1} \]
8. Step-by-step derivation

9. Applied rewrites 7.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))