normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 11.9s

Alternatives: 7

Speedup: 1.4×

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: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $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. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.6

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

5. Simplified 99.6%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (* (log u1) -2.0)) (cos (* 2.0 (* PI u2))))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((sqrt((log(u1) * -2.0)) * cos((2.0 * (((double) M_PI) * u2)))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(log(u1) * -2.0)) * cos(Float64(2.0 * Float64(pi * u2)))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(2.0 * N[(Pi * u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 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. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow1/2 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. metadata-eval 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (* (log u1) -2.0)) 0.16666666666666666)
  (cos (* 2.0 (* PI u2)))
  0.5))

C

double code(double u1, double u2) {
	return fma((sqrt((log(u1) * -2.0)) * 0.16666666666666666), cos((2.0 * (((double) M_PI) * u2))), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(log(u1) * -2.0)) * 0.16666666666666666), cos(Float64(2.0 * Float64(pi * u2))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Cos[N[(2.0 * N[(Pi * u2), $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. accelerator-lowering-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. unpow1/2 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. metadata-eval N/A

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

   9. cos-lowering-cos.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. PI-lowering-PI.f64 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 4: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(u2 * u2), $MachinePrecision] * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot -2}, 1\right) + \frac{1}{2} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot -2}, 1\right) + \frac{1}{2} \]

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. PI-lowering-PI.f64 N/A

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

   13. PI-lowering-PI.f64 98.7

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

   \[\leadsto 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right) \]
10. Add Preprocessing

Alternative 5: 98.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (* (log u1) -2.0)) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((sqrt((log(u1) * -2.0)) * fma(u2, (u2 * (-2.0 * (((double) M_PI) * ((double) M_PI)))), 1.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(log(u1) * -2.0)) * fma(u2, Float64(u2 * Float64(-2.0 * Float64(pi * pi))), 1.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[(u2 * N[(u2 * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 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. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot -2}, 1\right) + \frac{1}{2} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot -2}, 1\right) + \frac{1}{2} \]

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. PI-lowering-PI.f64 N/A

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

   13. PI-lowering-PI.f64 98.7

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

8. Simplified 98.7%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \left(\pi \cdot \pi\right) \cdot -2, 1\right)} + 0.5 \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) + 1\right) + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) + 1\right) + \frac{1}{2} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(u2 \cdot u2\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) + 1\right) \cdot \left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} + \frac{1}{2} \]

   4. accelerator-lowering-fma.f64 N/A

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

10. Applied egg-rr 98.6%

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

    1. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\left(\left(u2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \cdot \frac{1}{6}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1 \cdot 2\right)}} + \frac{1}{2} \]

    2. metadata-eval N/A

       \[\leadsto \left(\left(\left(u2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)}\right) \cdot \frac{1}{6}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1 \cdot 2\right)} + \frac{1}{2} \]

    3. sub-neg N/A

       \[\leadsto \left(\color{blue}{\left(\left(u2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot \mathsf{PI}\left(\right)\right)\right) - -1\right)} \cdot \frac{1}{6}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1 \cdot 2\right)} + \frac{1}{2} \]

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

       \[\leadsto \left(\left(\left(u2 \cdot u2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot \mathsf{PI}\left(\right)\right)\right) - -1\right) \cdot \frac{1}{6}\right) \cdot \sqrt{\color{blue}{\log u1 \cdot \left(\mathsf{neg}\left(2\right)\right)}} + \frac{1}{2} \]

    5. metadata-eval N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. accelerator-lowering-fma.f64 N/A

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

12. Applied egg-rr 98.6%

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

13. Final simplification 98.6%

    \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), 0.16666666666666666, 0.5\right) \]
14. Add Preprocessing

Alternative 6: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{2} \cdot 0.16666666666666666, \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (* (sqrt 2.0) 0.16666666666666666) (sqrt (- (log u1))) 0.5))

C

double code(double u1, double u2) {
	return fma((sqrt(2.0) * 0.16666666666666666), sqrt(-log(u1)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(2.0) * 0.16666666666666666), sqrt(Float64(-log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $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. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 0.0

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

5. Simplified 0.0%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

   \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot 0.16666666666666666, \sqrt{-\log u1}, 0.5\right) \]
9. Add Preprocessing

Alternative 7: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -2.0)) 0.16666666666666666 0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -2.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -2.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666 + 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. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 0.0

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

5. Simplified 0.0%

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

   1. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -2}} \cdot \frac{1}{6} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \log u1}} \cdot \frac{1}{6} + \frac{1}{2} \]

   5. unpow1/2 N/A

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

   6. metadata-eval N/A

      \[\leadsto {\left(-2 \cdot \log u1\right)}^{\color{blue}{\left(\frac{1}{4} \cdot 2\right)}} \cdot \frac{1}{6} + \frac{1}{2} \]

   7. pow-pow N/A

      \[\leadsto \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{\frac{1}{4}}\right)}^{2}} \cdot \frac{1}{6} + \frac{1}{2} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({\left(-2 \cdot \log u1\right)}^{\frac{1}{4}}\right)}^{2}, \frac{1}{6}, \frac{1}{2}\right)} \]

   9. pow-pow N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(-2 \cdot \log u1\right)}^{\left(\frac{1}{4} \cdot 2\right)}}, \frac{1}{6}, \frac{1}{2}\right) \]

   10. metadata-eval N/A

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

   11. unpow1/2 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 N/A

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

   15. log-lowering-log.f64 N/A

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

   16. metadata-eval 97.9

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

7. Applied egg-rr 97.9%

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

Reproduce

herbie shell --seed 2024205 
(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))