exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 17.2s

Alternatives: 8

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: 99.4% 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: 99.4% 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 99.8%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.4% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 97.3% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.8%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]
6. Add Preprocessing

Alternative 4: 76.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -1.1666666666666667\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (+ l (* l (* w (+ (* w (+ 0.5 (* w -1.1666666666666667))) -1.0)))))

C

double code(double w, double l) {
	return l + (l * (w * ((w * (0.5 + (w * -1.1666666666666667))) + -1.0)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (l * (w * ((w * (0.5d0 + (w * (-1.1666666666666667d0)))) + (-1.0d0))))
end function

Java

public static double code(double w, double l) {
	return l + (l * (w * ((w * (0.5 + (w * -1.1666666666666667))) + -1.0)));
}

Python

def code(w, l):
	return l + (l * (w * ((w * (0.5 + (w * -1.1666666666666667))) + -1.0)))

Julia

function code(w, l)
	return Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -1.1666666666666667))) + -1.0))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (l * (w * ((w * (0.5 + (w * -1.1666666666666667))) + -1.0)));
end

Wolfram

code[w_, l_] := N[(l + N[(l * N[(w * N[(N[(w * N[(0.5 + N[(w * -1.1666666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.8%

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

6. Taylor expanded in w around 0 76.7%

   \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
7. Step-by-step derivation

   1. associate-+r+ 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   2. mul-1-neg 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\color{blue}{\left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   3. distribute-rgt-out 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   4. metadata-eval 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\left(-\ell \cdot \color{blue}{-0.5}\right) + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   5. distribute-rgt-neg-in 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\color{blue}{\ell \cdot \left(--0.5\right)} + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   6. metadata-eval 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot \color{blue}{0.5} + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   7. add-sqr-sqrt 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   8. sqrt-unprod 56.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\sqrt{\ell \cdot \ell}}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. sqr-neg 56.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \sqrt{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   10. mul-1-neg 56.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \sqrt{\color{blue}{\left(-1 \cdot \ell\right)} \cdot \left(-\ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   11. mul-1-neg 56.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \sqrt{\left(-1 \cdot \ell\right) \cdot \color{blue}{\left(-1 \cdot \ell\right)}}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   12. sqrt-unprod 0.0%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   13. add-sqr-sqrt 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(-1 \cdot \ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   14. mul-1-neg 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(-\ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   15. distribute-rgt-neg-in 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \color{blue}{\left(--0.5 \cdot \ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   16. *-commutative 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \left(-\color{blue}{\ell \cdot -0.5}\right)\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   17. distribute-rgt-neg-in 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \color{blue}{\ell \cdot \left(--0.5\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   18. metadata-eval 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \ell \cdot \color{blue}{0.5}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   19. *-commutative 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \ell \cdot 0.5\right) + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

8. Applied egg-rr 76.7%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\left(\ell \cdot 0.5 + \ell \cdot 0.5\right) + \ell \cdot 0.16666666666666666\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
9. Step-by-step derivation

   1. distribute-lft-out 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\ell \cdot \left(0.5 + 0.5\right)} + \ell \cdot 0.16666666666666666\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   2. metadata-eval 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{1} + \ell \cdot 0.16666666666666666\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   3. *-rgt-identity 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\ell} + \ell \cdot 0.16666666666666666\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

10. Simplified 76.7%

    \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell + \ell \cdot 0.16666666666666666\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

11. Taylor expanded in l around 0 78.8%

    \[\leadsto \ell + \color{blue}{\ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + -1.1666666666666667 \cdot w\right) - 1\right)\right)} \]

12. Final simplification 78.8%

    \[\leadsto \ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -1.1666666666666667\right) + -1\right)\right) \]
13. Add Preprocessing

Alternative 5: 69.6% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* w (- (* w (* l 0.5)) l))))

C

double code(double w, double l) {
	return l + (w * ((w * (l * 0.5)) - l));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * ((w * (l * 0.5d0)) - l))
end function

Java

public static double code(double w, double l) {
	return l + (w * ((w * (l * 0.5)) - l));
}

Python

def code(w, l):
	return l + (w * ((w * (l * 0.5)) - l))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(Float64(w * Float64(l * 0.5)) - l)))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * ((w * (l * 0.5)) - l));
end

Wolfram

code[w_, l_] := N[(l + N[(w * N[(N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.8%

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

6. Taylor expanded in w around 0 76.7%

   \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
7. Step-by-step derivation

   1. associate-+r+ 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   2. mul-1-neg 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\color{blue}{\left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   3. distribute-rgt-out 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   4. metadata-eval 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\left(-\ell \cdot \color{blue}{-0.5}\right) + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   5. distribute-rgt-neg-in 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\color{blue}{\ell \cdot \left(--0.5\right)} + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   6. metadata-eval 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot \color{blue}{0.5} + -0.5 \cdot \ell\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   7. add-sqr-sqrt 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   8. sqrt-unprod 56.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\sqrt{\ell \cdot \ell}}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. sqr-neg 56.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \sqrt{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   10. mul-1-neg 56.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \sqrt{\color{blue}{\left(-1 \cdot \ell\right)} \cdot \left(-\ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   11. mul-1-neg 56.8%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \sqrt{\left(-1 \cdot \ell\right) \cdot \color{blue}{\left(-1 \cdot \ell\right)}}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   12. sqrt-unprod 0.0%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   13. add-sqr-sqrt 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(-1 \cdot \ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   14. mul-1-neg 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + -0.5 \cdot \color{blue}{\left(-\ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   15. distribute-rgt-neg-in 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \color{blue}{\left(--0.5 \cdot \ell\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   16. *-commutative 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \left(-\color{blue}{\ell \cdot -0.5}\right)\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   17. distribute-rgt-neg-in 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \color{blue}{\ell \cdot \left(--0.5\right)}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   18. metadata-eval 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \ell \cdot \color{blue}{0.5}\right) + 0.16666666666666666 \cdot \ell\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   19. *-commutative 76.7%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\left(\ell \cdot 0.5 + \ell \cdot 0.5\right) + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

8. Applied egg-rr 76.7%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\left(\ell \cdot 0.5 + \ell \cdot 0.5\right) + \ell \cdot 0.16666666666666666\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
9. Step-by-step derivation

   1. distribute-lft-out 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\ell \cdot \left(0.5 + 0.5\right)} + \ell \cdot 0.16666666666666666\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   2. metadata-eval 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\ell \cdot \color{blue}{1} + \ell \cdot 0.16666666666666666\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   3. *-rgt-identity 76.7%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(\color{blue}{\ell} + \ell \cdot 0.16666666666666666\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

10. Simplified 76.7%

    \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell + \ell \cdot 0.16666666666666666\right)}\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

11. Taylor expanded in w around 0 72.6%

    \[\leadsto \ell + w \cdot \left(\color{blue}{-1 \cdot \left(w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell\right) \]
12. Step-by-step derivation

    1. mul-1-neg 72.6%

       \[\leadsto \ell + w \cdot \left(\color{blue}{\left(-w \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell\right) \]

    2. distribute-rgt-neg-in 72.6%

       \[\leadsto \ell + w \cdot \left(\color{blue}{w \cdot \left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} - \ell\right) \]

    3. distribute-rgt-out 72.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) - \ell\right) \]

    4. metadata-eval 72.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-\ell \cdot \color{blue}{-0.5}\right) - \ell\right) \]

    5. metadata-eval 72.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-\ell \cdot \color{blue}{\left(-0.5\right)}\right) - \ell\right) \]

    6. distribute-rgt-neg-in 72.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \color{blue}{\left(\ell \cdot \left(-\left(-0.5\right)\right)\right)} - \ell\right) \]

    7. metadata-eval 72.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(\ell \cdot \left(-\color{blue}{-0.5}\right)\right) - \ell\right) \]

    8. metadata-eval 72.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(\ell \cdot \color{blue}{0.5}\right) - \ell\right) \]

13. Simplified 72.6%

    \[\leadsto \ell + w \cdot \left(\color{blue}{w \cdot \left(\ell \cdot 0.5\right)} - \ell\right) \]
14. Add Preprocessing

Alternative 6: 63.5% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.9:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -3.9) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3.9) {
		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 <= (-3.9d0)) 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 <= -3.9) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3.9:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3.9)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -3.9)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3.9], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -3.89999999999999991

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 97.5%

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

   6. Taylor expanded in w around 0 23.3%

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

      1. mul-1-neg 23.3%

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

      2. unsub-neg 23.3%

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

   8. Simplified 23.3%

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

   9. Taylor expanded in w around inf 23.3%

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

       1. mul-1-neg 23.3%

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

       2. *-commutative 23.3%

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

       3. distribute-rgt-neg-in 23.3%

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

   11. Simplified 23.3%

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

   if -3.89999999999999991 < w

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 97.9%

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

   6. Taylor expanded in w around 0 82.2%

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

Alternative 7: 63.2% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.8%

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

6. Taylor expanded in w around 0 64.4%

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

   1. mul-1-neg 64.4%

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

   2. unsub-neg 64.4%

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

8. Simplified 64.4%

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

9. Final simplification 64.4%

   \[\leadsto \ell - w \cdot \ell \]
10. Add Preprocessing

Alternative 8: 56.8% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 97.8%

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

6. Taylor expanded in w around 0 58.8%

   \[\leadsto \color{blue}{\ell} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))