exp-w (used to crash)

Percentage Accurate: 99.5% → 99.4%

Time: 11.6s

Alternatives: 8

Speedup: N/A×

• 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: 99.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left(e^{w \cdot 0.5}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/ (pow (pow l (sqrt (exp w))) (exp (* w 0.5))) (exp w)))

C

double code(double w, double l) {
	return pow(pow(l, sqrt(exp(w))), exp((w * 0.5))) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l ** sqrt(exp(w))) ** exp((w * 0.5d0))) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow(Math.pow(l, Math.sqrt(Math.exp(w))), Math.exp((w * 0.5))) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(math.pow(l, math.sqrt(math.exp(w))), math.exp((w * 0.5))) / math.exp(w)

Julia

function code(w, l)
	return Float64(((l ^ sqrt(exp(w))) ^ exp(Float64(w * 0.5))) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l ^ sqrt(exp(w))) ^ exp((w * 0.5))) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[N[Power[l, N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Exp[N[(w * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 97.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{\left(e^{w}\right)}\right)\right)}}{e^{w}} \]

5. Applied egg-rr 97.4%

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\ell}^{\left(e^{w}\right)}\right)\right)}}{e^{w}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. pow-unpow 99.5%

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

7. Applied egg-rr 99.5%

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

   1. pow1/2 99.5%

      \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left({\left(e^{w}\right)}^{0.5}\right)}}}{e^{w}} \]

   2. pow-exp 99.5%

      \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left(e^{w \cdot 0.5}\right)}}}{e^{w}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\color{blue}{\left(e^{w \cdot 0.5}\right)}}}{e^{w}} \]

10. Final simplification 99.5%

    \[\leadsto \frac{{\left({\ell}^{\left(\sqrt{e^{w}}\right)}\right)}^{\left(e^{w \cdot 0.5}\right)}}{e^{w}} \]

Alternative 2: 99.5% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 97.7% accurate, 2.9× speedup

Math

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

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(Float64(-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 99.5%

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

2. Taylor expanded in w around 0 96.8%

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

3. Final simplification 96.8%

   \[\leadsto \ell \cdot e^{-w} \]

Alternative 4: 97.7% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.8%

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

5. Final simplification 96.8%

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

Alternative 5: 72.2% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.09:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell \cdot \left(w + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.09) (* l (- 1.0 w)) (/ (* l l) (* l (+ w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.09) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (l * l) / (l * (w + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.09d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = (l * l) / (l * (w + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.09) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (l * l) / (l * (w + 1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.09:
		tmp = l * (1.0 - w)
	else:
		tmp = (l * l) / (l * (w + 1.0))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.09)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(Float64(l * l) / Float64(l * Float64(w + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.09)
		tmp = l * (1.0 - w);
	else
		tmp = (l * l) / (l * (w + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.09], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(l * N[(w + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.089999999999999997

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. associate-*l/ 99.5%

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

      3. *-lft-identity 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in w around 0 97.2%

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

   5. Taylor expanded in w around 0 73.1%

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

      1. +-commutative 73.1%

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

      2. mul-1-neg 73.1%

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

      3. unsub-neg 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in l around 0 73.1%

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

   if 0.089999999999999997 < w

   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. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in w around 0 94.0%

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

   5. Taylor expanded in w around 0 3.3%

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

      1. +-commutative 3.3%

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

      2. mul-1-neg 3.3%

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

      3. unsub-neg 3.3%

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

   7. Simplified 3.3%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]
   8. Step-by-step derivation

      1. flip-- 6.3%

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

      2. difference-of-squares 6.3%

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

      3. add-sqr-sqrt 6.3%

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

      4. sqrt-unprod 5.6%

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

      5. sqr-neg 5.6%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 9.5%

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

      8. distribute-rgt-neg-in 9.5%

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

      9. sub-neg 9.5%

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

      10. pow2 9.5%

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

      11. sub-neg 9.5%

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

      12. *-commutative 9.5%

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

      13. distribute-lft-neg-in 9.5%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 8.8%

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

      16. sqr-neg 8.8%

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

      17. sqrt-unprod 9.5%

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

      18. add-sqr-sqrt 9.5%

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

      19. distribute-rgt1-in 9.5%

          \[\leadsto \frac{{\color{blue}{\left(\left(w + 1\right) \cdot \ell\right)}}^{2}}{\ell + \ell \cdot w} \]

      20. *-commutative 9.5%

          \[\leadsto \frac{{\left(\left(w + 1\right) \cdot \ell\right)}^{2}}{\ell + \color{blue}{w \cdot \ell}} \]

   9. Applied egg-rr 9.5%

      \[\leadsto \color{blue}{\frac{{\left(\left(w + 1\right) \cdot \ell\right)}^{2}}{\left(w + 1\right) \cdot \ell}} \]

   10. Taylor expanded in w around 0 67.1%

       \[\leadsto \frac{\color{blue}{{\ell}^{2}}}{\left(w + 1\right) \cdot \ell} \]
   11. Step-by-step derivation

       1. unpow2 67.1%

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

   12. Simplified 67.1%

       \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{\left(w + 1\right) \cdot \ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.09:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell \cdot \left(w + 1\right)}\\ \end{array} \]

Alternative 6: 63.4% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.8%

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

5. Taylor expanded in w around 0 64.6%

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

   1. +-commutative 64.6%

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

   2. mul-1-neg 64.6%

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

   3. unsub-neg 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in l around 0 64.6%

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

9. Final simplification 64.6%

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

Alternative 7: 10.8% accurate, 76.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.8%

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

5. Taylor expanded in w around 0 64.6%

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

   1. +-commutative 64.6%

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

   2. mul-1-neg 64.6%

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

   3. unsub-neg 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in w around inf 10.4%

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

   1. associate-*r* 10.4%

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

   2. neg-mul-1 10.4%

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

10. Simplified 10.4%

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

11. Final simplification 10.4%

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

Alternative 8: 3.2% accurate, 101.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.8%

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

5. Taylor expanded in w around 0 64.6%

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

   1. +-commutative 64.6%

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

   2. mul-1-neg 64.6%

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

   3. unsub-neg 64.6%

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

7. Simplified 64.6%

   \[\leadsto \color{blue}{\ell - \ell \cdot w} \]
8. Step-by-step derivation

   1. flip-- 34.7%

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

   2. difference-of-squares 36.7%

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

   3. add-sqr-sqrt 17.0%

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

   4. sqrt-unprod 27.6%

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

   5. sqr-neg 27.6%

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

   6. sqrt-unprod 10.8%

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

   7. add-sqr-sqrt 28.1%

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

   8. distribute-rgt-neg-in 28.1%

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

   9. sub-neg 28.1%

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

   10. pow2 28.1%

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

   11. sub-neg 28.1%

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

   12. *-commutative 28.1%

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

   13. distribute-lft-neg-in 28.1%

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

   14. add-sqr-sqrt 10.8%

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

   15. sqrt-unprod 28.0%

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

   16. sqr-neg 28.0%

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

   17. sqrt-unprod 17.3%

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

   18. add-sqr-sqrt 28.1%

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

   19. distribute-rgt1-in 28.1%

       \[\leadsto \frac{{\color{blue}{\left(\left(w + 1\right) \cdot \ell\right)}}^{2}}{\ell + \ell \cdot w} \]

   20. *-commutative 28.1%

       \[\leadsto \frac{{\left(\left(w + 1\right) \cdot \ell\right)}^{2}}{\ell + \color{blue}{w \cdot \ell}} \]

9. Applied egg-rr 28.1%

   \[\leadsto \color{blue}{\frac{{\left(\left(w + 1\right) \cdot \ell\right)}^{2}}{\left(w + 1\right) \cdot \ell}} \]

10. Taylor expanded in w around inf 3.4%

    \[\leadsto \color{blue}{\ell \cdot w} \]
11. Step-by-step derivation

    1. *-commutative 3.4%

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

12. Simplified 3.4%

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

13. Final simplification 3.4%

    \[\leadsto \ell \cdot w \]

Reproduce

herbie shell --seed 2023213 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))