normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 13.3s

Alternatives: 3

Speedup: 1.0×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log u1 \cdot -0.05555555555555555} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (* (sqrt (* (log u1) -0.05555555555555555)) (cos (* (* 2.0 PI) u2))) 0.5))

C

double code(double u1, double u2) {
	return (sqrt((log(u1) * -0.05555555555555555)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (Math.sqrt((Math.log(u1) * -0.05555555555555555)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (math.sqrt((math.log(u1) * -0.05555555555555555)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(sqrt(Float64(log(u1) * -0.05555555555555555)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (sqrt((log(u1) * -0.05555555555555555)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 99.4%

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. add-sqr-sqrt 99.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}} \cdot \sqrt{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. sqrt-unprod 99.4%

      \[\leadsto \color{blue}{\sqrt{\left(\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \left(\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   4. *-commutative 99.4%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right)} \cdot \left(\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. *-commutative 99.4%

      \[\leadsto \sqrt{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right) \cdot \color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   6. swap-sqr 99.4%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   7. add-sqr-sqrt 99.6%

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot \log u1\right)} \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   8. metadata-eval 99.6%

      \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(\color{blue}{0.16666666666666666} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   9. metadata-eval 99.6%

      \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(0.16666666666666666 \cdot \color{blue}{0.16666666666666666}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   10. metadata-eval 99.6%

       \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \color{blue}{0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. associate-*l* 99.6%

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. metadata-eval 99.6%

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
7. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \sqrt{-1 + \mathsf{fma}\left(-0.05555555555555555, \log u1, 1\right)} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ 0.5 (sqrt (+ -1.0 (fma -0.05555555555555555 (log u1) 1.0)))))

C

double code(double u1, double u2) {
	return 0.5 + sqrt((-1.0 + fma(-0.05555555555555555, log(u1), 1.0)));
}

Julia

function code(u1, u2)
	return Float64(0.5 + sqrt(Float64(-1.0 + fma(-0.05555555555555555, log(u1), 1.0))))
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Sqrt[N[(-1.0 + N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot \left(\frac{1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} + 0.5 \]

   3. fma-define 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.5}, \frac{1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)} \]

   4. unpow1/2 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{-2 \cdot \log u1}}, \frac{1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

   5. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{0.16666666666666666} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

   6. associate-*l* 99.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}, 0.5\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in u2 around 0 98.0%

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{0.16666666666666666}, 0.5\right) \]
6. Step-by-step derivation

   1. fma-undefine 98.0%

      \[\leadsto \color{blue}{\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666 + 0.5} \]

   2. *-commutative 98.0%

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot -2}} \cdot 0.16666666666666666 + 0.5 \]

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666 + 0.5} \]
8. Step-by-step derivation

   1. pow1 98.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666\right)}^{1}} + 0.5 \]

9. Applied egg-rr 51.0%

   \[\leadsto \color{blue}{{\left(\sqrt{\log \left({u1}^{-2}\right) \cdot 0.027777777777777776}\right)}^{1}} + 0.5 \]
10. Step-by-step derivation

    1. unpow1 51.0%

       \[\leadsto \color{blue}{\sqrt{\log \left({u1}^{-2}\right) \cdot 0.027777777777777776}} + 0.5 \]

11. Simplified 51.0%

    \[\leadsto \color{blue}{\sqrt{\log \left({u1}^{-2}\right) \cdot 0.027777777777777776}} + 0.5 \]
12. Step-by-step derivation

    1. pow-to-exp 51.0%

       \[\leadsto \sqrt{\log \color{blue}{\left(e^{\log u1 \cdot -2}\right)} \cdot 0.027777777777777776} + 0.5 \]

    2. add-log-exp 98.3%

       \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot 0.027777777777777776} + 0.5 \]

    3. associate-*l* 98.3%

       \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} + 0.5 \]

    4. metadata-eval 98.3%

       \[\leadsto \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} + 0.5 \]

    5. expm1-log1p-u 98.0%

       \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log u1 \cdot -0.05555555555555555\right)\right)}} + 0.5 \]

    6. expm1-undefine 98.0%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\log u1 \cdot -0.05555555555555555\right)} - 1}} + 0.5 \]

13. Applied egg-rr 98.0%

    \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\log u1 \cdot -0.05555555555555555\right)} - 1}} + 0.5 \]
14. Step-by-step derivation

    1. sub-neg 98.0%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\log u1 \cdot -0.05555555555555555\right)} + \left(-1\right)}} + 0.5 \]

    2. log1p-undefine 98.0%

       \[\leadsto \sqrt{e^{\color{blue}{\log \left(1 + \log u1 \cdot -0.05555555555555555\right)}} + \left(-1\right)} + 0.5 \]

    3. rem-exp-log 98.3%

       \[\leadsto \sqrt{\color{blue}{\left(1 + \log u1 \cdot -0.05555555555555555\right)} + \left(-1\right)} + 0.5 \]

    4. metadata-eval 98.3%

       \[\leadsto \sqrt{\left(1 + \log u1 \cdot -0.05555555555555555\right) + \color{blue}{-1}} + 0.5 \]

    5. +-commutative 98.3%

       \[\leadsto \sqrt{\color{blue}{-1 + \left(1 + \log u1 \cdot -0.05555555555555555\right)}} + 0.5 \]

    6. +-commutative 98.3%

       \[\leadsto \sqrt{-1 + \color{blue}{\left(\log u1 \cdot -0.05555555555555555 + 1\right)}} + 0.5 \]

    7. *-commutative 98.3%

       \[\leadsto \sqrt{-1 + \left(\color{blue}{-0.05555555555555555 \cdot \log u1} + 1\right)} + 0.5 \]

    8. fma-define 98.3%

       \[\leadsto \sqrt{-1 + \color{blue}{\mathsf{fma}\left(-0.05555555555555555, \log u1, 1\right)}} + 0.5 \]

15. Simplified 98.3%

    \[\leadsto \sqrt{\color{blue}{-1 + \mathsf{fma}\left(-0.05555555555555555, \log u1, 1\right)}} + 0.5 \]

16. Final simplification 98.3%

    \[\leadsto 0.5 + \sqrt{-1 + \mathsf{fma}\left(-0.05555555555555555, \log u1, 1\right)} \]
17. Add Preprocessing

Alternative 3: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log u1 \cdot -0.05555555555555555} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (* (log u1) -0.05555555555555555)) 0.5))

C

double code(double u1, double u2) {
	return sqrt((log(u1) * -0.05555555555555555)) + 0.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((log(u1) * (-0.05555555555555555d0))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((Math.log(u1) * -0.05555555555555555)) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt((math.log(u1) * -0.05555555555555555)) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(log(u1) * -0.05555555555555555)) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((log(u1) * -0.05555555555555555)) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot \left(\frac{1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} + 0.5 \]

   3. fma-define 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.5}, \frac{1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)} \]

   4. unpow1/2 99.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{-2 \cdot \log u1}}, \frac{1}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

   5. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{0.16666666666666666} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]

   6. associate-*l* 99.3%

      \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}, 0.5\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in u2 around 0 98.0%

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{0.16666666666666666}, 0.5\right) \]
6. Step-by-step derivation

   1. pow1/2 98.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(-2 \cdot \log u1\right)}^{0.5}}, 0.16666666666666666, 0.5\right) \]

7. Applied egg-rr 98.0%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(-2 \cdot \log u1\right)}^{0.5}}, 0.16666666666666666, 0.5\right) \]
8. Step-by-step derivation

   1. pow1/2 98.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{-2 \cdot \log u1}}, 0.16666666666666666, 0.5\right) \]

   2. add-cbrt-cube 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{\left(\sqrt{-2 \cdot \log u1} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \sqrt{-2 \cdot \log u1}}}, 0.16666666666666666, 0.5\right) \]

   3. add-sqr-sqrt 97.8%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{\left(-2 \cdot \log u1\right)} \cdot \sqrt{-2 \cdot \log u1}}, 0.16666666666666666, 0.5\right) \]

   4. pow1 97.8%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{{\left(-2 \cdot \log u1\right)}^{1}} \cdot \sqrt{-2 \cdot \log u1}}, 0.16666666666666666, 0.5\right) \]

   5. pow1/2 97.8%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(-2 \cdot \log u1\right)}^{1} \cdot \color{blue}{{\left(-2 \cdot \log u1\right)}^{0.5}}}, 0.16666666666666666, 0.5\right) \]

   6. pow-prod-up 97.8%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{{\left(-2 \cdot \log u1\right)}^{\left(1 + 0.5\right)}}}, 0.16666666666666666, 0.5\right) \]

   7. metadata-eval 97.8%

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(-2 \cdot \log u1\right)}^{\color{blue}{1.5}}}, 0.16666666666666666, 0.5\right) \]

9. Applied egg-rr 97.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{{\left(-2 \cdot \log u1\right)}^{1.5}}}, 0.16666666666666666, 0.5\right) \]
10. Step-by-step derivation

    1. pow1/3 96.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{1.5}\right)}^{0.3333333333333333}}, 0.16666666666666666, 0.5\right) \]

    2. log-pow 50.4%

       \[\leadsto \mathsf{fma}\left({\left({\color{blue}{\log \left({u1}^{-2}\right)}}^{1.5}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

    3. pow-pow 50.9%

       \[\leadsto \mathsf{fma}\left(\color{blue}{{\log \left({u1}^{-2}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}, 0.16666666666666666, 0.5\right) \]

    4. pow-to-exp 50.9%

       \[\leadsto \mathsf{fma}\left({\log \color{blue}{\left(e^{\log u1 \cdot -2}\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}, 0.16666666666666666, 0.5\right) \]

    5. add-log-exp 98.0%

       \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\log u1 \cdot -2\right)}}^{\left(1.5 \cdot 0.3333333333333333\right)}, 0.16666666666666666, 0.5\right) \]

    6. rem-cube-cbrt 97.4%

       \[\leadsto \mathsf{fma}\left({\left(\color{blue}{{\left(\sqrt[3]{\log u1}\right)}^{3}} \cdot -2\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}, 0.16666666666666666, 0.5\right) \]

    7. metadata-eval 97.4%

       \[\leadsto \mathsf{fma}\left({\left({\left(\sqrt[3]{\log u1}\right)}^{3} \cdot -2\right)}^{\color{blue}{0.5}}, 0.16666666666666666, 0.5\right) \]

    8. pow1/2 97.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{{\left(\sqrt[3]{\log u1}\right)}^{3} \cdot -2}}, 0.16666666666666666, 0.5\right) \]

    9. fma-define 97.4%

       \[\leadsto \color{blue}{\sqrt{{\left(\sqrt[3]{\log u1}\right)}^{3} \cdot -2} \cdot 0.16666666666666666 + 0.5} \]

    10. *-commutative 97.4%

        \[\leadsto \color{blue}{0.16666666666666666 \cdot \sqrt{{\left(\sqrt[3]{\log u1}\right)}^{3} \cdot -2}} + 0.5 \]

11. Applied egg-rr 98.3%

    \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555} + 0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))