exp-w (used to crash)

Percentage Accurate: 99.5% → 98.7%

Time: 15.0s

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: 99.5% 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: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(\log \ell + -1\right)\\ \mathbf{if}\;w \leq -0.22:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.48:\\ \;\;\;\;\ell + \ell \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* w (+ (log l) -1.0))))
   (if (<= w -0.22) (exp (- w)) (if (<= w 0.48) (+ l (* l t_0)) (exp t_0)))))

C

double code(double w, double l) {
	double t_0 = w * (log(l) + -1.0);
	double tmp;
	if (w <= -0.22) {
		tmp = exp(-w);
	} else if (w <= 0.48) {
		tmp = l + (l * t_0);
	} else {
		tmp = exp(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w * (log(l) + (-1.0d0))
    if (w <= (-0.22d0)) then
        tmp = exp(-w)
    else if (w <= 0.48d0) then
        tmp = l + (l * t_0)
    else
        tmp = exp(t_0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = w * (Math.log(l) + -1.0);
	double tmp;
	if (w <= -0.22) {
		tmp = Math.exp(-w);
	} else if (w <= 0.48) {
		tmp = l + (l * t_0);
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = w * (math.log(l) + -1.0)
	tmp = 0
	if w <= -0.22:
		tmp = math.exp(-w)
	elif w <= 0.48:
		tmp = l + (l * t_0)
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(w, l)
	t_0 = Float64(w * Float64(log(l) + -1.0))
	tmp = 0.0
	if (w <= -0.22)
		tmp = exp(Float64(-w));
	elseif (w <= 0.48)
		tmp = Float64(l + Float64(l * t_0));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = w * (log(l) + -1.0);
	tmp = 0.0;
	if (w <= -0.22)
		tmp = exp(-w);
	elseif (w <= 0.48)
		tmp = l + (l * t_0);
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -0.22], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 0.48], N[(l + N[(l * t$95$0), $MachinePrecision]), $MachinePrecision], N[Exp[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if w < -0.220000000000000001

   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 l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -0.220000000000000001 < w < 0.47999999999999998

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. fma-define 98.2%

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

      3. sub-neg 98.2%

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

      4. *-commutative 98.2%

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

      5. neg-mul-1 98.2%

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

      6. distribute-rgt-out 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in w around 0 98.2%

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

   if 0.47999999999999998 < w

   1. Initial program 95.7%

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

      1. exp-neg 95.7%

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

      2. remove-double-neg 95.7%

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

      3. associate-*l/ 95.7%

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

      4. *-lft-identity 95.7%

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

      5. remove-double-neg 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in l around inf 95.7%

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

      1. mul-1-neg 95.7%

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

      2. distribute-rgt-neg-in 95.7%

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

      3. log-rec 95.7%

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

      4. remove-double-neg 95.7%

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

      5. +-rgt-identity 95.7%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around inf 100.0%

       \[\leadsto e^{\color{blue}{w \cdot \left(\log \ell - 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.22:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.48:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{w \cdot \left(\log \ell + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow-unpow 98.9%

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

   3. pow-to-exp 94.6%

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

6. Applied egg-rr 94.6%

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

   1. exp-to-pow 98.9%

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

8. Simplified 98.9%

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

   1. pow1/2 98.9%

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

   2. pow-exp 98.9%

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

10. Applied egg-rr 98.9%

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

Alternative 3: 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 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

Alternative 4: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -52000000:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \left(\ell \cdot w\right) \cdot \log \ell}{e^{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -52000000.0) (exp (- w)) (/ (+ l (* (* l w) (log l))) (exp w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -52000000.0) {
		tmp = exp(-w);
	} else {
		tmp = (l + ((l * w) * log(l))) / exp(w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-52000000.0d0)) then
        tmp = exp(-w)
    else
        tmp = (l + ((l * w) * log(l))) / exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -52000000.0) {
		tmp = Math.exp(-w);
	} else {
		tmp = (l + ((l * w) * Math.log(l))) / Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -52000000.0:
		tmp = math.exp(-w)
	else:
		tmp = (l + ((l * w) * math.log(l))) / math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -52000000.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64(Float64(l + Float64(Float64(l * w) * log(l))) / exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -52000000.0)
		tmp = exp(-w);
	else
		tmp = (l + ((l * w) * log(l))) / exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -52000000.0], N[Exp[(-w)], $MachinePrecision], N[(N[(l + N[(N[(l * w), $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -5.2e7

   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 l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -5.2e7 < w

   1. Initial program 98.4%

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

      1. exp-neg 98.4%

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

      2. remove-double-neg 98.4%

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

      3. associate-*l/ 98.4%

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

      4. *-lft-identity 98.4%

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

      5. remove-double-neg 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in w around 0 97.0%

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

      1. associate-*r* 97.5%

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

   7. Simplified 97.5%

      \[\leadsto \frac{\color{blue}{\ell + \left(\ell \cdot w\right) \cdot \log \ell}}{e^{w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -52000000:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(w + 1\right)}}{e^{w}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -52000000.0) (exp (- w)) (/ (pow l (+ w 1.0)) (exp w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -52000000.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (w + 1.0)) / exp(w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-52000000.0d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** (w + 1.0d0)) / exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -52000000.0) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (w + 1.0)) / Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -52000000.0:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, (w + 1.0)) / math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -52000000.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ Float64(w + 1.0)) / exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -52000000.0)
		tmp = exp(-w);
	else
		tmp = (l ^ (w + 1.0)) / exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -52000000.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -5.2e7

   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 l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -5.2e7 < w

   1. Initial program 98.4%

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

      1. exp-neg 98.4%

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

      2. remove-double-neg 98.4%

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

      3. associate-*l/ 98.4%

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

      4. *-lft-identity 98.4%

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

      5. remove-double-neg 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in l around inf 92.0%

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

      1. mul-1-neg 92.0%

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

      2. distribute-rgt-neg-in 92.0%

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

      3. log-rec 92.0%

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

      4. remove-double-neg 92.0%

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

      5. +-rgt-identity 92.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 93.2%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 93.2%

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

      8. remove-double-neg 93.2%

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

      9. distribute-lft-neg-in 93.2%

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

      10. distribute-rgt-neg-out 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in w around 0 92.3%

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

      1. distribute-rgt1-in 92.3%

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

   10. Simplified 92.3%

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

   11. Taylor expanded in w around inf 92.3%

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

       1. exp-diff 91.1%

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

       2. exp-to-pow 97.5%

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

       3. +-commutative 97.5%

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

   13. Simplified 97.5%

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

Alternative 6: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.19 \lor \neg \left(w \leq 3900000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.19) (not (<= w 3900000.0)))
   (exp (- w))
   (+ l (* l (* w (+ (log l) -1.0))))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.19) || !(w <= 3900000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l + (l * (w * (log(l) + -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 <= (-0.19d0)) .or. (.not. (w <= 3900000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l + (l * (w * (log(l) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.19) || !(w <= 3900000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l + (l * (w * (Math.log(l) + -1.0)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.19) or not (w <= 3900000.0):
		tmp = math.exp(-w)
	else:
		tmp = l + (l * (w * (math.log(l) + -1.0)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.19) || !(w <= 3900000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l + Float64(l * Float64(w * Float64(log(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.19) || ~((w <= 3900000.0)))
		tmp = exp(-w);
	else
		tmp = l + (l * (w * (log(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.19], N[Not[LessEqual[w, 3900000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l + N[(l * N[(w * N[(N[Log[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.19 or 3.9e6 < 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. 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 l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -0.19 < w < 3.9e6

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in w around 0 96.8%

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

      1. +-commutative 96.8%

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

      2. fma-define 96.8%

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

      3. sub-neg 96.8%

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

      4. *-commutative 96.8%

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

      5. neg-mul-1 96.8%

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

      6. distribute-rgt-out 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in w around 0 96.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.19 \lor \neg \left(w \leq 3900000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(\log \ell + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 3900000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 3900000.0))) (exp (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 3900000.0)) {
		tmp = exp(-w);
	} 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 <= (-0.7d0)) .or. (.not. (w <= 3900000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 3900000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 3900000.0):
		tmp = math.exp(-w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 3900000.0))
		tmp = exp(Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 3900000.0)))
		tmp = exp(-w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 3900000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 3.9e6 < 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. 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 l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -0.69999999999999996 < w < 3.9e6

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. remove-double-neg 97.9%

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

      3. associate-*l/ 97.9%

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

      4. *-lft-identity 97.9%

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

      5. remove-double-neg 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in w around 0 95.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 3900000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.6% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 98.9%

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

   1. exp-neg 98.9%

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

   2. remove-double-neg 98.9%

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

   3. associate-*l/ 98.9%

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

   4. *-lft-identity 98.9%

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

   5. remove-double-neg 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in w around 0 51.2%

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

Reproduce

herbie shell --seed 2024088 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))