exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 12.9s

Alternatives: 12

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{{t_0}^{-2} \cdot {\ell}^{\left(e^{w}\right)}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (exp w)))) (/ (* (pow t_0 -2.0) (pow l (exp w))) t_0)))

C

double code(double w, double l) {
	double t_0 = cbrt(exp(w));
	return (pow(t_0, -2.0) * pow(l, exp(w))) / t_0;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.cbrt(Math.exp(w));
	return (Math.pow(t_0, -2.0) * Math.pow(l, Math.exp(w))) / t_0;
}

Julia

function code(w, l)
	t_0 = cbrt(exp(w))
	return Float64(Float64((t_0 ^ -2.0) * (l ^ exp(w))) / t_0)
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[Power[t$95$0, -2.0], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.3%

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

   2. unpow-prod-down 99.3%

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

5. Applied egg-rr 99.3%

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

   1. pow-sqr 99.3%

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

   2. *-commutative 99.3%

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

7. Simplified 99.3%

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

   1. pow-unpow 99.3%

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

   2. unpow2 99.3%

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

   3. unpow-prod-down 99.3%

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

   4. add-sqr-sqrt 99.7%

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

   5. *-un-lft-identity 99.7%

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

   6. add-cube-cbrt 99.7%

      \[\leadsto \frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   7. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

   8. pow2 99.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
10. Step-by-step derivation

    1. associate-*r/ 99.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot {\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

    2. pow-flip 99.7%

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

    3. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

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(Float64(-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 99.7%

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

2. Final simplification 99.7%

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

Alternative 3: 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 97.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 700\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 700.0))) (exp (- w)) (+ l (* w l))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 700.0)) {
		tmp = exp(-w);
	} else {
		tmp = l + (w * 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 <= (-0.7d0)) .or. (.not. (w <= 700.0d0))) then
        tmp = exp(-w)
    else
        tmp = l + (w * l)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 700.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l + (w * l);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 700.0):
		tmp = math.exp(-w)
	else:
		tmp = l + (w * l)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 700.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l + Float64(w * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 700.0)))
		tmp = exp(-w);
	else
		tmp = l + (w * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 700.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l + N[(w * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 700 < w

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. unpow-prod-down 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. pow-sqr 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. pow-unpow 100.0%

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

      2. unpow2 100.0%

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

      3. unpow-prod-down 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. expm1-log1p-u 100.0%

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

      6. expm1-udef 100.0%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{\left(e^{w}\right)}\right)} - 1}}{e^{w}} \]

      7. log1p-udef 100.0%

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

      8. add-exp-log 100.0%

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

      9. add-exp-log 99.9%

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

      10. log-div 99.9%

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

      11. add-exp-log 99.9%

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

      12. log1p-udef 99.9%

          \[\leadsto e^{\log \left(e^{\color{blue}{\mathsf{log1p}\left({\ell}^{\left(e^{w}\right)}\right)}} - 1\right) - \log \left(e^{w}\right)} \]

      13. expm1-udef 99.9%

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

      14. expm1-log1p-u 99.9%

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

      15. log-pow 99.9%

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

      16. add-log-exp 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in w around inf 99.6%

       \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   11. Step-by-step derivation

       1. neg-mul-1 99.6%

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

   12. Simplified 99.6%

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

   if -0.69999999999999996 < w < 700

   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 97.3%

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

   5. Taylor expanded in w around 0 97.3%

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

      1. +-commutative 97.3%

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

      2. mul-1-neg 97.3%

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

      3. unsub-neg 97.3%

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

   7. Simplified 97.3%

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

      1. sub-neg 97.3%

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

      2. flip-+ 48.5%

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

      3. distribute-rgt-neg-in 48.5%

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

      4. distribute-rgt-neg-in 48.5%

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

      5. distribute-rgt-neg-in 48.5%

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

   9. Applied egg-rr 48.5%

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

       1. flip-+ 97.3%

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

       2. *-commutative 97.3%

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

       3. add-sqr-sqrt 45.6%

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

       4. sqrt-unprod 97.3%

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

       5. sqr-neg 97.3%

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

       6. sqrt-unprod 51.7%

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

       7. add-sqr-sqrt 97.3%

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

       8. cancel-sign-sub 97.3%

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

       9. *-commutative 97.3%

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

       10. *-un-lft-identity 97.3%

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

       11. sub-neg 97.3%

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

       12. distribute-rgt-in 97.3%

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

       13. add-sqr-sqrt 96.6%

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

       14. sqrt-prod 49.0%

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

       15. sqr-neg 49.0%

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

       16. sqrt-unprod 0.0%

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

       17. add-sqr-sqrt 2.5%

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

       18. *-rgt-identity 2.5%

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

       19. add-sqr-sqrt 0.0%

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

       20. sqrt-unprod 49.0%

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

       21. sqr-neg 49.0%

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

       22. sqrt-prod 96.6%

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

       23. add-sqr-sqrt 97.3%

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

   11. Applied egg-rr 97.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 700\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \ell\\ \end{array} \]

Alternative 5: 97.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \frac{1}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (/ 1.0 (exp w))))

C

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

Fortran

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

Java

public static double code(double w, double l) {
	return l * (1.0 / Math.exp(w));
}

Python

def code(w, l):
	return l * (1.0 / math.exp(w))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 98.1%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]
5. Step-by-step derivation

   1. frac-2neg 98.1%

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

   2. div-inv 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in w around inf 98.1%

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

8. Final simplification 98.1%

   \[\leadsto \ell \cdot \frac{1}{e^{w}} \]

Alternative 6: 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 98.1%

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

5. Final simplification 98.1%

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

Alternative 7: 82.7% accurate, 20.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.065:\\ \;\;\;\;\left(\ell - w \cdot \ell\right) - \left(\ell \cdot -0.5\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell + w \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.065)
   (- (- l (* w l)) (* (* l -0.5) (* w w)))
   (/ (* l l) (+ l (* w l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.065) {
		tmp = (l - (w * l)) - ((l * -0.5) * (w * w));
	} else {
		tmp = (l * l) / (l + (w * 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 <= 0.065d0) then
        tmp = (l - (w * l)) - ((l * (-0.5d0)) * (w * w))
    else
        tmp = (l * l) / (l + (w * l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.065) {
		tmp = (l - (w * l)) - ((l * -0.5) * (w * w));
	} else {
		tmp = (l * l) / (l + (w * l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.065:
		tmp = (l - (w * l)) - ((l * -0.5) * (w * w))
	else:
		tmp = (l * l) / (l + (w * l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.065)
		tmp = Float64(Float64(l - Float64(w * l)) - Float64(Float64(l * -0.5) * Float64(w * w)));
	else
		tmp = Float64(Float64(l * l) / Float64(l + Float64(w * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.065)
		tmp = (l - (w * l)) - ((l * -0.5) * (w * w));
	else
		tmp = (l * l) / (l + (w * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.065], N[(N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision] - N[(N[(l * -0.5), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(l + N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.065000000000000002

   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. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

   3. Simplified 99.7%

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

      1. expm1-log1p-u 97.2%

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

      2. expm1-udef 63.1%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left({\ell}^{\left(e^{w}\right)}\right)} - 1}}{e^{w}} \]

      3. log1p-udef 63.1%

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

      4. add-exp-log 65.5%

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

   5. Applied egg-rr 65.5%

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

   6. Taylor expanded in w around 0 49.0%

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

   7. Taylor expanded in w around 0 85.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot w\right) + \left(\ell + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 85.7%

         \[\leadsto \color{blue}{\left(-\ell \cdot w\right)} + \left(\ell + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right)\right) \]

      2. distribute-rgt-neg-out 85.7%

         \[\leadsto \color{blue}{\ell \cdot \left(-w\right)} + \left(\ell + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right)\right) \]

      3. associate-+r+ 85.7%

         \[\leadsto \color{blue}{\left(\ell \cdot \left(-w\right) + \ell\right) + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right)} \]

      4. +-commutative 85.7%

         \[\leadsto \color{blue}{\left(\ell + \ell \cdot \left(-w\right)\right)} + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right) \]

      5. *-commutative 85.7%

         \[\leadsto \left(\ell + \color{blue}{\left(-w\right) \cdot \ell}\right) + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right) \]

      6. cancel-sign-sub-inv 85.7%

         \[\leadsto \color{blue}{\left(\ell - w \cdot \ell\right)} + -1 \cdot \left(\left(-1 \cdot \ell + 0.5 \cdot \ell\right) \cdot {w}^{2}\right) \]

      7. associate-*r* 85.7%

         \[\leadsto \left(\ell - w \cdot \ell\right) + \color{blue}{\left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) \cdot {w}^{2}} \]

      8. mul-1-neg 85.7%

         \[\leadsto \left(\ell - w \cdot \ell\right) + \color{blue}{\left(-\left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} \cdot {w}^{2} \]

      9. distribute-rgt-out 85.7%

         \[\leadsto \left(\ell - w \cdot \ell\right) + \left(-\color{blue}{\ell \cdot \left(-1 + 0.5\right)}\right) \cdot {w}^{2} \]

      10. metadata-eval 85.7%

          \[\leadsto \left(\ell - w \cdot \ell\right) + \left(-\ell \cdot \color{blue}{-0.5}\right) \cdot {w}^{2} \]

      11. unpow2 85.7%

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

   9. Simplified 85.7%

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

   if 0.065000000000000002 < w

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 97.7%

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

   5. Taylor expanded in w around 0 3.4%

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

      1. +-commutative 3.4%

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

      2. mul-1-neg 3.4%

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

      3. unsub-neg 3.4%

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

   7. Simplified 3.4%

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

      1. sub-neg 3.4%

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

      2. flip-+ 19.4%

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

      3. distribute-rgt-neg-in 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. distribute-rgt-neg-in 19.4%

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

   9. Applied egg-rr 19.4%

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

   10. Taylor expanded in w around 0 65.6%

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

       1. unpow2 65.6%

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

   12. Simplified 65.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.065:\\ \;\;\;\;\left(\ell - w \cdot \ell\right) - \left(\ell \cdot -0.5\right) \cdot \left(w \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell + w \cdot \ell}\\ \end{array} \]

Alternative 8: 75.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.052:\\ \;\;\;\;\frac{1 - w \cdot w}{\frac{w + 1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell + w \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.052)
   (/ (- 1.0 (* w w)) (/ (+ w 1.0) l))
   (/ (* l l) (+ l (* w l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.052) {
		tmp = (1.0 - (w * w)) / ((w + 1.0) / l);
	} else {
		tmp = (l * l) / (l + (w * 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 <= 0.052d0) then
        tmp = (1.0d0 - (w * w)) / ((w + 1.0d0) / l)
    else
        tmp = (l * l) / (l + (w * l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.052) {
		tmp = (1.0 - (w * w)) / ((w + 1.0) / l);
	} else {
		tmp = (l * l) / (l + (w * l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.052:
		tmp = (1.0 - (w * w)) / ((w + 1.0) / l)
	else:
		tmp = (l * l) / (l + (w * l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.052)
		tmp = Float64(Float64(1.0 - Float64(w * w)) / Float64(Float64(w + 1.0) / l));
	else
		tmp = Float64(Float64(l * l) / Float64(l + Float64(w * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.052)
		tmp = (1.0 - (w * w)) / ((w + 1.0) / l);
	else
		tmp = (l * l) / (l + (w * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.052], N[(N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(N[(w + 1.0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(l + N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.0519999999999999976

   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. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in w around 0 98.2%

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

   5. Taylor expanded in w around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

   7. Simplified 74.6%

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

      1. sub-neg 74.6%

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

      2. flip-+ 37.8%

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

      3. distribute-rgt-neg-in 37.8%

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

      4. distribute-rgt-neg-in 37.8%

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

      5. distribute-rgt-neg-in 37.8%

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

   9. Applied egg-rr 37.8%

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

   10. Taylor expanded in l around 0 85.6%

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

       1. associate-/l* 78.9%

          \[\leadsto \color{blue}{\frac{1 - {w}^{2}}{\frac{1 - -1 \cdot w}{\ell}}} \]

       2. unpow2 78.9%

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

       3. sub-neg 78.9%

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

       4. mul-1-neg 78.9%

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

       5. remove-double-neg 78.9%

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

   12. Simplified 78.9%

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

   if 0.0519999999999999976 < w

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 97.7%

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

   5. Taylor expanded in w around 0 3.4%

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

      1. +-commutative 3.4%

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

      2. mul-1-neg 3.4%

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

      3. unsub-neg 3.4%

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

   7. Simplified 3.4%

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

      1. sub-neg 3.4%

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

      2. flip-+ 19.4%

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

      3. distribute-rgt-neg-in 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. distribute-rgt-neg-in 19.4%

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

   9. Applied egg-rr 19.4%

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

   10. Taylor expanded in w around 0 65.6%

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

       1. unpow2 65.6%

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

   12. Simplified 65.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.052:\\ \;\;\;\;\frac{1 - w \cdot w}{\frac{w + 1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell + w \cdot \ell}\\ \end{array} \]

Alternative 9: 72.8% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.125:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell + w \cdot \ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.125) (* l (- 1.0 w)) (/ (* l l) (+ l (* w l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.125) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (l * l) / (l + (w * 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 <= 0.125d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = (l * l) / (l + (w * l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.125) {
		tmp = l * (1.0 - w);
	} else {
		tmp = (l * l) / (l + (w * l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.125:
		tmp = l * (1.0 - w)
	else:
		tmp = (l * l) / (l + (w * l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.125)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(Float64(l * l) / Float64(l + Float64(w * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.125)
		tmp = l * (1.0 - w);
	else
		tmp = (l * l) / (l + (w * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.125], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] / N[(l + N[(w * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.125

   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. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in w around 0 98.2%

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

   5. Taylor expanded in w around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in l around 0 74.6%

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

   if 0.125 < w

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 97.7%

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

   5. Taylor expanded in w around 0 3.4%

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

      1. +-commutative 3.4%

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

      2. mul-1-neg 3.4%

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

      3. unsub-neg 3.4%

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

   7. Simplified 3.4%

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

      1. sub-neg 3.4%

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

      2. flip-+ 19.4%

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

      3. distribute-rgt-neg-in 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. distribute-rgt-neg-in 19.4%

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

   9. Applied egg-rr 19.4%

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

   10. Taylor expanded in w around 0 65.6%

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

       1. unpow2 65.6%

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

   12. Simplified 65.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.125:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{\ell + w \cdot \ell}\\ \end{array} \]

Alternative 10: 64.4% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.64:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.64) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.64) {
		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 <= (-0.64d0)) 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 <= -0.64) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.64:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.64)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.64)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.64], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.640000000000000013

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in w around 0 98.7%

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

   5. Taylor expanded in w around 0 30.1%

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

      1. +-commutative 30.1%

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

      2. mul-1-neg 30.1%

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

      3. unsub-neg 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in w around inf 30.1%

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

      1. mul-1-neg 30.1%

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

      2. distribute-rgt-neg-out 30.1%

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

   10. Simplified 30.1%

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

   if -0.640000000000000013 < w

   1. Initial program 99.6%

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

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

      1. add-sqr-sqrt 99.1%

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

      2. unpow-prod-down 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. pow-sqr 99.1%

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

      2. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. pow-unpow 99.1%

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

      2. unpow2 99.1%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. add-cube-cbrt 99.6%

         \[\leadsto \frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

      7. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

      8. pow2 99.6%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot {\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

       2. pow-flip 99.6%

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

       3. metadata-eval 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in w around 0 77.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.64:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 11: 64.1% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 98.1%

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

5. Taylor expanded in w around 0 63.2%

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

   1. +-commutative 63.2%

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

   2. mul-1-neg 63.2%

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

   3. unsub-neg 63.2%

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

7. Simplified 63.2%

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

8. Taylor expanded in l around 0 63.2%

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

9. Final simplification 63.2%

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

Alternative 12: 57.4% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.3%

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

   2. unpow-prod-down 99.3%

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

5. Applied egg-rr 99.3%

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

   1. pow-sqr 99.3%

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

   2. *-commutative 99.3%

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

7. Simplified 99.3%

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

   1. pow-unpow 99.3%

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

   2. unpow2 99.3%

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

   3. unpow-prod-down 99.3%

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

   4. add-sqr-sqrt 99.7%

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

   5. *-un-lft-identity 99.7%

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

   6. add-cube-cbrt 99.7%

      \[\leadsto \frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   7. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

   8. pow2 99.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
10. Step-by-step derivation

    1. associate-*r/ 99.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot {\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]

    2. pow-flip 99.7%

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

    3. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

12. Taylor expanded in w around 0 56.0%

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

13. Final simplification 56.0%

    \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023224 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))