exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 15.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. w = 2.00000000000000007e154
  2. l = -2

Specification

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Final simplification 98.9%

   \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (exp w)))

C

double code(double w, double l) {
	return pow(l, exp(w)) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (l ** exp(w)) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow(l, Math.exp(w)) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(l, math.exp(w)) / math.exp(w)

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 98.9%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 98.9%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 98.9%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Final simplification 98.9%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]

Alternative 3: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\sqrt[3]{e^{w}}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (cbrt (exp w))))

C

double code(double w, double l) {
	return l / cbrt(exp(w));
}

Java

public static double code(double w, double l) {
	return l / Math.cbrt(Math.exp(w));
}

Julia

function code(w, l)
	return Float64(l / cbrt(exp(w)))
end

Wolfram

code[w_, l_] := N[(l / N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. *-commutative 98.9%

      \[\leadsto \color{blue}{{\ell}^{\left(e^{w}\right)} \cdot e^{-w}} \]

   2. exp-neg 98.9%

      \[\leadsto {\ell}^{\left(e^{w}\right)} \cdot \color{blue}{\frac{1}{e^{w}}} \]

   3. div-inv 98.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   4. add-cube-cbrt 98.9%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   5. associate-/r* 98.9%

      \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}{\sqrt[3]{e^{w}}}} \]

   6. pow2 98.9%

      \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}}}{\sqrt[3]{e^{w}}} \]

3. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}}{\sqrt[3]{e^{w}}}} \]

4. Taylor expanded in w around 0 97.3%

   \[\leadsto \frac{\color{blue}{\ell}}{\sqrt[3]{e^{w}}} \]

5. Final simplification 97.3%

   \[\leadsto \frac{\ell}{\sqrt[3]{e^{w}}} \]

Alternative 4: 97.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (exp w)))

C

double code(double w, double l) {
	return l / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l / exp(w)
end function

Java

public static double code(double w, double l) {
	return l / Math.exp(w);
}

Python

def code(w, l):
	return l / math.exp(w)

Julia

function code(w, l)
	return Float64(l / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = l / exp(w);
end

Wolfram

code[w_, l_] := N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 98.9%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 98.9%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 98.9%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 98.9%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

      \[\leadsto \frac{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{\left(e^{w}\right)}}{e^{w}} \]

   2. unpow-prod-down 98.5%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}}}{e^{w}} \]

5. Applied egg-rr 98.5%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)} \cdot {\left(\sqrt{\ell}\right)}^{\left(e^{w}\right)}}}{e^{w}} \]
6. Step-by-step derivation

   1. pow-sqr 98.5%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{\left(2 \cdot e^{w}\right)}}}{e^{w}} \]

7. Simplified 98.5%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{\left(2 \cdot e^{w}\right)}}}{e^{w}} \]

8. Taylor expanded in w around 0 97.3%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

9. Final simplification 97.3%

   \[\leadsto \frac{\ell}{e^{w}} \]

Alternative 5: 68.6% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \left(1 - w \cdot w\right) \cdot \frac{\ell}{w + 1} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (- 1.0 (* w w)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	return (1.0 - (w * w)) * (l / (w + 1.0));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (1.0d0 - (w * w)) * (l / (w + 1.0d0))
end function

Java

public static double code(double w, double l) {
	return (1.0 - (w * w)) * (l / (w + 1.0));
}

Python

def code(w, l):
	return (1.0 - (w * w)) * (l / (w + 1.0))

Julia

function code(w, l)
	return Float64(Float64(1.0 - Float64(w * w)) * Float64(l / Float64(w + 1.0)))
end

MATLAB

function tmp = code(w, l)
	tmp = (1.0 - (w * w)) * (l / (w + 1.0));
end

Wolfram

code[w_, l_] := N[(N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Taylor expanded in w around 0 58.4%

   \[\leadsto \color{blue}{\ell + \left(-1 \cdot \ell + \ell \cdot \log \ell\right) \cdot w} \]
3. Step-by-step derivation

   1. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]

   2. mul-1-neg 58.4%

      \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-\ell\right)} + \ell \cdot \log \ell\right) \]

   3. +-commutative 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell + \left(-\ell\right)\right)} \]

   4. sub-neg 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell - \ell\right)} \]

4. Simplified 58.4%

   \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

5. Taylor expanded in l around 0 58.7%

   \[\leadsto \ell + \color{blue}{\left(\log \ell - 1\right) \cdot \left(\ell \cdot w\right)} \]
6. Step-by-step derivation

   1. associate-*r* 58.4%

      \[\leadsto \ell + \color{blue}{\left(\left(\log \ell - 1\right) \cdot \ell\right) \cdot w} \]

   2. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \left(\log \ell - 1\right)\right)} \cdot w \]

   3. distribute-lft-out-- 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \log \ell - \ell \cdot 1\right)} \cdot w \]

   4. log-pow 32.5%

      \[\leadsto \ell + \left(\color{blue}{\log \left({\ell}^{\ell}\right)} - \ell \cdot 1\right) \cdot w \]

   5. *-rgt-identity 32.5%

      \[\leadsto \ell + \left(\log \left({\ell}^{\ell}\right) - \color{blue}{\ell}\right) \cdot w \]

   6. *-commutative 32.5%

      \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

7. Simplified 32.5%

   \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

8. Taylor expanded in l around inf 65.4%

   \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 65.4%

      \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

   2. unsub-neg 65.4%

      \[\leadsto \ell \cdot \color{blue}{\left(1 - w\right)} \]

10. Simplified 65.4%

    \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]
11. Step-by-step derivation

    1. flip-- 72.7%

       \[\leadsto \ell \cdot \color{blue}{\frac{1 \cdot 1 - w \cdot w}{1 + w}} \]

    2. associate-*r/ 74.9%

       \[\leadsto \color{blue}{\frac{\ell \cdot \left(1 \cdot 1 - w \cdot w\right)}{1 + w}} \]

    3. metadata-eval 74.9%

       \[\leadsto \frac{\ell \cdot \left(\color{blue}{1} - w \cdot w\right)}{1 + w} \]

    4. +-commutative 74.9%

       \[\leadsto \frac{\ell \cdot \left(1 - w \cdot w\right)}{\color{blue}{w + 1}} \]

12. Applied egg-rr 74.9%

    \[\leadsto \color{blue}{\frac{\ell \cdot \left(1 - w \cdot w\right)}{w + 1}} \]
13. Step-by-step derivation

    1. associate-/l* 72.7%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{w + 1}{1 - w \cdot w}}} \]

    2. associate-/r/ 69.9%

       \[\leadsto \color{blue}{\frac{\ell}{w + 1} \cdot \left(1 - w \cdot w\right)} \]

14. Simplified 69.9%

    \[\leadsto \color{blue}{\frac{\ell}{w + 1} \cdot \left(1 - w \cdot w\right)} \]

15. Final simplification 69.9%

    \[\leadsto \left(1 - w \cdot w\right) \cdot \frac{\ell}{w + 1} \]

Alternative 6: 72.2% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\frac{w + 1}{1 - w \cdot w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (/ (+ w 1.0) (- 1.0 (* w w)))))

C

double code(double w, double l) {
	return l / ((w + 1.0) / (1.0 - (w * w)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l / ((w + 1.0d0) / (1.0d0 - (w * w)))
end function

Java

public static double code(double w, double l) {
	return l / ((w + 1.0) / (1.0 - (w * w)));
}

Python

def code(w, l):
	return l / ((w + 1.0) / (1.0 - (w * w)))

Julia

function code(w, l)
	return Float64(l / Float64(Float64(w + 1.0) / Float64(1.0 - Float64(w * w))))
end

MATLAB

function tmp = code(w, l)
	tmp = l / ((w + 1.0) / (1.0 - (w * w)));
end

Wolfram

code[w_, l_] := N[(l / N[(N[(w + 1.0), $MachinePrecision] / N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Taylor expanded in w around 0 58.4%

   \[\leadsto \color{blue}{\ell + \left(-1 \cdot \ell + \ell \cdot \log \ell\right) \cdot w} \]
3. Step-by-step derivation

   1. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]

   2. mul-1-neg 58.4%

      \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-\ell\right)} + \ell \cdot \log \ell\right) \]

   3. +-commutative 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell + \left(-\ell\right)\right)} \]

   4. sub-neg 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell - \ell\right)} \]

4. Simplified 58.4%

   \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

5. Taylor expanded in l around 0 58.7%

   \[\leadsto \ell + \color{blue}{\left(\log \ell - 1\right) \cdot \left(\ell \cdot w\right)} \]
6. Step-by-step derivation

   1. associate-*r* 58.4%

      \[\leadsto \ell + \color{blue}{\left(\left(\log \ell - 1\right) \cdot \ell\right) \cdot w} \]

   2. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \left(\log \ell - 1\right)\right)} \cdot w \]

   3. distribute-lft-out-- 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \log \ell - \ell \cdot 1\right)} \cdot w \]

   4. log-pow 32.5%

      \[\leadsto \ell + \left(\color{blue}{\log \left({\ell}^{\ell}\right)} - \ell \cdot 1\right) \cdot w \]

   5. *-rgt-identity 32.5%

      \[\leadsto \ell + \left(\log \left({\ell}^{\ell}\right) - \color{blue}{\ell}\right) \cdot w \]

   6. *-commutative 32.5%

      \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

7. Simplified 32.5%

   \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

8. Taylor expanded in l around inf 65.4%

   \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 65.4%

      \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

   2. unsub-neg 65.4%

      \[\leadsto \ell \cdot \color{blue}{\left(1 - w\right)} \]

10. Simplified 65.4%

    \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]
11. Step-by-step derivation

    1. flip-- 72.7%

       \[\leadsto \ell \cdot \color{blue}{\frac{1 \cdot 1 - w \cdot w}{1 + w}} \]

    2. associate-*r/ 74.9%

       \[\leadsto \color{blue}{\frac{\ell \cdot \left(1 \cdot 1 - w \cdot w\right)}{1 + w}} \]

    3. metadata-eval 74.9%

       \[\leadsto \frac{\ell \cdot \left(\color{blue}{1} - w \cdot w\right)}{1 + w} \]

    4. +-commutative 74.9%

       \[\leadsto \frac{\ell \cdot \left(1 - w \cdot w\right)}{\color{blue}{w + 1}} \]

12. Applied egg-rr 74.9%

    \[\leadsto \color{blue}{\frac{\ell \cdot \left(1 - w \cdot w\right)}{w + 1}} \]
13. Step-by-step derivation

    1. associate-/l* 72.7%

       \[\leadsto \color{blue}{\frac{\ell}{\frac{w + 1}{1 - w \cdot w}}} \]

14. Simplified 72.7%

    \[\leadsto \color{blue}{\frac{\ell}{\frac{w + 1}{1 - w \cdot w}}} \]

15. Final simplification 72.7%

    \[\leadsto \frac{\ell}{\frac{w + 1}{1 - w \cdot w}} \]

Alternative 7: 73.9% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot \left(1 - w \cdot w\right)}{w + 1} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (* l (- 1.0 (* w w))) (+ w 1.0)))

C

double code(double w, double l) {
	return (l * (1.0 - (w * w))) / (w + 1.0);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (l * (1.0d0 - (w * w))) / (w + 1.0d0)
end function

Java

public static double code(double w, double l) {
	return (l * (1.0 - (w * w))) / (w + 1.0);
}

Python

def code(w, l):
	return (l * (1.0 - (w * w))) / (w + 1.0)

Julia

function code(w, l)
	return Float64(Float64(l * Float64(1.0 - Float64(w * w))) / Float64(w + 1.0))
end

MATLAB

function tmp = code(w, l)
	tmp = (l * (1.0 - (w * w))) / (w + 1.0);
end

Wolfram

code[w_, l_] := N[(N[(l * N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Taylor expanded in w around 0 58.4%

   \[\leadsto \color{blue}{\ell + \left(-1 \cdot \ell + \ell \cdot \log \ell\right) \cdot w} \]
3. Step-by-step derivation

   1. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]

   2. mul-1-neg 58.4%

      \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-\ell\right)} + \ell \cdot \log \ell\right) \]

   3. +-commutative 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell + \left(-\ell\right)\right)} \]

   4. sub-neg 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell - \ell\right)} \]

4. Simplified 58.4%

   \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

5. Taylor expanded in l around 0 58.7%

   \[\leadsto \ell + \color{blue}{\left(\log \ell - 1\right) \cdot \left(\ell \cdot w\right)} \]
6. Step-by-step derivation

   1. associate-*r* 58.4%

      \[\leadsto \ell + \color{blue}{\left(\left(\log \ell - 1\right) \cdot \ell\right) \cdot w} \]

   2. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \left(\log \ell - 1\right)\right)} \cdot w \]

   3. distribute-lft-out-- 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \log \ell - \ell \cdot 1\right)} \cdot w \]

   4. log-pow 32.5%

      \[\leadsto \ell + \left(\color{blue}{\log \left({\ell}^{\ell}\right)} - \ell \cdot 1\right) \cdot w \]

   5. *-rgt-identity 32.5%

      \[\leadsto \ell + \left(\log \left({\ell}^{\ell}\right) - \color{blue}{\ell}\right) \cdot w \]

   6. *-commutative 32.5%

      \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

7. Simplified 32.5%

   \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

8. Taylor expanded in l around inf 65.4%

   \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 65.4%

      \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

   2. unsub-neg 65.4%

      \[\leadsto \ell \cdot \color{blue}{\left(1 - w\right)} \]

10. Simplified 65.4%

    \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]
11. Step-by-step derivation

    1. *-commutative 65.4%

       \[\leadsto \color{blue}{\left(1 - w\right) \cdot \ell} \]

    2. flip-- 72.7%

       \[\leadsto \color{blue}{\frac{1 \cdot 1 - w \cdot w}{1 + w}} \cdot \ell \]

    3. associate-*l/ 74.9%

       \[\leadsto \color{blue}{\frac{\left(1 \cdot 1 - w \cdot w\right) \cdot \ell}{1 + w}} \]

    4. metadata-eval 74.9%

       \[\leadsto \frac{\left(\color{blue}{1} - w \cdot w\right) \cdot \ell}{1 + w} \]

    5. +-commutative 74.9%

       \[\leadsto \frac{\left(1 - w \cdot w\right) \cdot \ell}{\color{blue}{w + 1}} \]

12. Applied egg-rr 74.9%

    \[\leadsto \color{blue}{\frac{\left(1 - w \cdot w\right) \cdot \ell}{w + 1}} \]

13. Final simplification 74.9%

    \[\leadsto \frac{\ell \cdot \left(1 - w \cdot w\right)}{w + 1} \]

Alternative 8: 64.1% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.0085:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.0085) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.0085) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.0085d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.0085) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.0085:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.0085)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.0085)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.0085], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.0085000000000000006

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. Taylor expanded in w around 0 2.0%

      \[\leadsto \color{blue}{\ell + \left(-1 \cdot \ell + \ell \cdot \log \ell\right) \cdot w} \]
   3. Step-by-step derivation

      1. *-commutative 2.0%

         \[\leadsto \ell + \color{blue}{w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]

      2. mul-1-neg 2.0%

         \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-\ell\right)} + \ell \cdot \log \ell\right) \]

      3. +-commutative 2.0%

         \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell + \left(-\ell\right)\right)} \]

      4. sub-neg 2.0%

         \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell - \ell\right)} \]

   4. Simplified 2.0%

      \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

   5. Taylor expanded in l around 0 2.0%

      \[\leadsto \ell + \color{blue}{\left(\log \ell - 1\right) \cdot \left(\ell \cdot w\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 2.0%

         \[\leadsto \ell + \color{blue}{\left(\left(\log \ell - 1\right) \cdot \ell\right) \cdot w} \]

      2. *-commutative 2.0%

         \[\leadsto \ell + \color{blue}{\left(\ell \cdot \left(\log \ell - 1\right)\right)} \cdot w \]

      3. distribute-lft-out-- 2.0%

         \[\leadsto \ell + \color{blue}{\left(\ell \cdot \log \ell - \ell \cdot 1\right)} \cdot w \]

      4. log-pow 1.9%

         \[\leadsto \ell + \left(\color{blue}{\log \left({\ell}^{\ell}\right)} - \ell \cdot 1\right) \cdot w \]

      5. *-rgt-identity 1.9%

         \[\leadsto \ell + \left(\log \left({\ell}^{\ell}\right) - \color{blue}{\ell}\right) \cdot w \]

      6. *-commutative 1.9%

         \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

   7. Simplified 1.9%

      \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

   8. Taylor expanded in l around inf 27.9%

      \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
   9. Step-by-step derivation

      1. neg-mul-1 27.9%

         \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

      2. unsub-neg 27.9%

         \[\leadsto \ell \cdot \color{blue}{\left(1 - w\right)} \]

   10. Simplified 27.9%

       \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]

   11. Taylor expanded in w around inf 27.9%

       \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot w\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 27.9%

          \[\leadsto \color{blue}{-\ell \cdot w} \]

       2. *-commutative 27.9%

          \[\leadsto -\color{blue}{w \cdot \ell} \]

       3. distribute-rgt-neg-in 27.9%

          \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   13. Simplified 27.9%

       \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   if -0.0085000000000000006 < w

   1. Initial program 98.5%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. Taylor expanded in w around 0 79.9%

      \[\leadsto \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.0085:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 9: 63.8% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 - w\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (- 1.0 w)))

C

double code(double w, double l) {
	return l * (1.0 - w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 - w)
end function

Java

public static double code(double w, double l) {
	return l * (1.0 - w);
}

Python

def code(w, l):
	return l * (1.0 - w)

Julia

function code(w, l)
	return Float64(l * Float64(1.0 - w))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 - w);
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Taylor expanded in w around 0 58.4%

   \[\leadsto \color{blue}{\ell + \left(-1 \cdot \ell + \ell \cdot \log \ell\right) \cdot w} \]
3. Step-by-step derivation

   1. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]

   2. mul-1-neg 58.4%

      \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-\ell\right)} + \ell \cdot \log \ell\right) \]

   3. +-commutative 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell + \left(-\ell\right)\right)} \]

   4. sub-neg 58.4%

      \[\leadsto \ell + w \cdot \color{blue}{\left(\ell \cdot \log \ell - \ell\right)} \]

4. Simplified 58.4%

   \[\leadsto \color{blue}{\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)} \]

5. Taylor expanded in l around 0 58.7%

   \[\leadsto \ell + \color{blue}{\left(\log \ell - 1\right) \cdot \left(\ell \cdot w\right)} \]
6. Step-by-step derivation

   1. associate-*r* 58.4%

      \[\leadsto \ell + \color{blue}{\left(\left(\log \ell - 1\right) \cdot \ell\right) \cdot w} \]

   2. *-commutative 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \left(\log \ell - 1\right)\right)} \cdot w \]

   3. distribute-lft-out-- 58.4%

      \[\leadsto \ell + \color{blue}{\left(\ell \cdot \log \ell - \ell \cdot 1\right)} \cdot w \]

   4. log-pow 32.5%

      \[\leadsto \ell + \left(\color{blue}{\log \left({\ell}^{\ell}\right)} - \ell \cdot 1\right) \cdot w \]

   5. *-rgt-identity 32.5%

      \[\leadsto \ell + \left(\log \left({\ell}^{\ell}\right) - \color{blue}{\ell}\right) \cdot w \]

   6. *-commutative 32.5%

      \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

7. Simplified 32.5%

   \[\leadsto \ell + \color{blue}{w \cdot \left(\log \left({\ell}^{\ell}\right) - \ell\right)} \]

8. Taylor expanded in l around inf 65.4%

   \[\leadsto \color{blue}{\ell \cdot \left(1 + -1 \cdot w\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 65.4%

      \[\leadsto \ell \cdot \left(1 + \color{blue}{\left(-w\right)}\right) \]

   2. unsub-neg 65.4%

      \[\leadsto \ell \cdot \color{blue}{\left(1 - w\right)} \]

10. Simplified 65.4%

    \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]

11. Final simplification 65.4%

    \[\leadsto \ell \cdot \left(1 - w\right) \]

Alternative 10: 57.5% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ \ell \end{array} \]

FPCore

(FPCore (w l) :precision binary64 l)

C

double code(double w, double l) {
	return l;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l
end function

Java

public static double code(double w, double l) {
	return l;
}

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

function tmp = code(w, l)
	tmp = l;
end

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 98.9%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]

2. Taylor expanded in w around 0 59.2%

   \[\leadsto \color{blue}{\ell} \]

3. Final simplification 59.2%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023252 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))