normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 13.6s

Alternatives: 5

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: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. pow1/2 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate-*r* 99.5%

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

   6. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-2 \cdot \log u1} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) + 0.5\right)\right)} \]

   7. fma-undefine 98.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)}\right)\right) \]

   8. expm1-undefine 98.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\right)} - 1} \]

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-2 \cdot \log u1}, 0.5\right)\right)} - 1} \]
5. Step-by-step derivation

   1. log1p-undefine 98.6%

      \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-2 \cdot \log u1}, 0.5\right)\right)}} - 1 \]

   2. rem-exp-log 99.3%

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

   3. +-commutative 99.3%

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

   4. associate--l+ 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \mathsf{fma}\left(0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-2 \cdot \log u1}, 0.5\right) + \color{blue}{0} \]

   6. +-rgt-identity 99.4%

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

   7. fma-undefine 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.5%

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

   10. fma-undefine 99.5%

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

6. Simplified 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right)} \]
7. Step-by-step derivation

   1. add-log-exp 99.7%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log \left(e^{\log u1 \cdot -0.05555555555555555}\right)}}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right) \]

   2. exp-to-pow 99.7%

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

8. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log \left({u1}^{-0.05555555555555555}\right)}}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right) \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. pow1/2 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate-*r* 99.5%

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

   6. expm1-log1p-u 98.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-2 \cdot \log u1} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) + 0.5\right)\right)} \]

   7. fma-undefine 98.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)}\right)\right) \]

   8. expm1-undefine 98.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\right)} - 1} \]

4. Applied egg-rr 98.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-2 \cdot \log u1}, 0.5\right)\right)} - 1} \]
5. Step-by-step derivation

   1. log1p-undefine 98.6%

      \[\leadsto e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-2 \cdot \log u1}, 0.5\right)\right)}} - 1 \]

   2. rem-exp-log 99.3%

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

   3. +-commutative 99.3%

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

   4. associate--l+ 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \mathsf{fma}\left(0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-2 \cdot \log u1}, 0.5\right) + \color{blue}{0} \]

   6. +-rgt-identity 99.4%

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

   7. fma-undefine 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.5%

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

   10. fma-undefine 99.5%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right) \]
8. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. expm1-undefine 99.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} - 1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. metadata-eval 99.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{0.16666666666666666} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} - 1\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   4. pow1/2 99.2%

      \[\leadsto \left(e^{\mathsf{log1p}\left(0.16666666666666666 \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right)} - 1\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} - 1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. log1p-undefine 99.2%

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

   2. rem-exp-log 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.5%

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

   5. metadata-eval 99.5%

      \[\leadsto \left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1} + \color{blue}{0}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   6. +-rgt-identity 99.5%

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

   7. rem-square-sqrt 99.0%

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

   8. fabs-sqr 99.0%

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

   9. rem-square-sqrt 99.5%

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

   10. rem-sqrt-square 99.5%

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

   11. swap-sqr 99.4%

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

   12. metadata-eval 99.4%

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

   13. rem-square-sqrt 99.7%

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

   14. *-commutative 99.7%

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

   15. *-commutative 99.7%

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

   16. associate-*l* 99.7%

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

   17. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(u1, u2)
	return Float64(sqrt(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[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.5%

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

   3. fma-define 99.5%

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

   4. unpow1/2 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u2 around 0 97.8%

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

   1. fma-undefine 97.8%

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

   2. *-commutative 97.8%

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

7. Applied egg-rr 97.8%

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

   1. add-sqr-sqrt 97.4%

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

   2. sqrt-unprod 97.8%

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

   3. *-commutative 97.8%

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

   4. *-commutative 97.8%

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

   5. swap-sqr 97.8%

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

   6. add-sqr-sqrt 98.0%

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

   7. *-commutative 98.0%

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

   8. metadata-eval 98.0%

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

9. Applied egg-rr 98.0%

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

    1. associate-*l* 98.0%

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

    2. metadata-eval 98.0%

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

11. Simplified 98.0%

    \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555}} + 0.5 \]
12. Step-by-step derivation

    1. add-log-exp 99.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log \left(e^{\log u1 \cdot -0.05555555555555555}\right)}}, \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.5\right) \]

    2. exp-to-pow 99.7%

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

13. Applied egg-rr 98.0%

    \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{-0.05555555555555555}\right)}} + 0.5 \]
14. Add Preprocessing

Alternative 5: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.5%

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

   3. fma-define 99.5%

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

   4. unpow1/2 99.5%

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

   5. metadata-eval 99.5%

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

   6. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u2 around 0 97.8%

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

   1. fma-undefine 97.8%

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

   2. *-commutative 97.8%

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

7. Applied egg-rr 97.8%

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

   1. add-sqr-sqrt 97.4%

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

   2. sqrt-unprod 97.8%

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

   3. *-commutative 97.8%

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

   4. *-commutative 97.8%

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

   5. swap-sqr 97.8%

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

   6. add-sqr-sqrt 98.0%

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

   7. *-commutative 98.0%

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

   8. metadata-eval 98.0%

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

9. Applied egg-rr 98.0%

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

    1. associate-*l* 98.0%

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

    2. metadata-eval 98.0%

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2024110 
(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))