normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 8.9s

Alternatives: 3

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left({\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}^{2} \cdot \left(\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776\right)\right)}^{0.5} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (pow
   (*
    (pow (cos (* (* 2.0 (PI)) u2)) 2.0)
    (* (* -2.0 (log u1)) 0.027777777777777776))
   0.5)
  0.5))

Derivation

1. Initial program 99.5%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 99.2

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 99.2%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(\log u1 \cdot -2\right)}^{0.25}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

   \[\leadsto {\left({\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}^{2} \cdot \left(\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776\right)\right)}^{0.5} + 0.5 \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\log u1 \cdot -2\right)}^{0.25}, {\left(\log u1 \cdot -2\right)}^{0.25} \cdot \left(\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot 0.16666666666666666\right), 0.5\right)} \]

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 98.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-log.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.6%

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

Reproduce

herbie shell --seed 2024276 
(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))