exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 12.2s

Alternatives: 9

Speedup: N/A×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 720\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 720.0))) (exp (- w)) (* l (exp w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 720.0)) {
		tmp = exp(-w);
	} else {
		tmp = 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 <= (-0.68d0)) .or. (.not. (w <= 720.0d0))) then
        tmp = exp(-w)
    else
        tmp = l * exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 720.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l * Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 720.0):
		tmp = math.exp(-w)
	else:
		tmp = l * math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 720.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l * exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 720.0)))
		tmp = exp(-w);
	else
		tmp = l * exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 720.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l * N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 720 < 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-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. div-exp 100.0%

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

      2. associate-*r* 100.0%

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

      3. neg-mul-1 100.0%

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

      4. fma-neg 100.0%

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

      5. log-rec 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   if -0.680000000000000049 < w < 720

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. associate-*l/ 99.2%

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

      3. *-lft-identity 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in w around 0 93.9%

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

      1. clear-num 93.6%

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

      2. associate-/r/ 93.9%

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

      3. exp-neg 93.9%

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

      4. add-sqr-sqrt 45.9%

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

      5. sqrt-unprod 94.6%

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

      6. sqr-neg 94.6%

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

      7. sqrt-unprod 48.7%

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

      8. add-sqr-sqrt 94.6%

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

   6. Applied egg-rr 94.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 720\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \]

Alternative 3: 97.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 720\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 720.0)))
   (exp (- w))
   (- l (+ (* l w) (* (* w w) (* l -0.5))))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 720.0)) {
		tmp = exp(-w);
	} else {
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.68d0)) .or. (.not. (w <= 720.0d0))) then
        tmp = exp(-w)
    else
        tmp = l - ((l * w) + ((w * w) * (l * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 720.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 720.0):
		tmp = math.exp(-w)
	else:
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 720.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l - Float64(Float64(l * w) + Float64(Float64(w * w) * Float64(l * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 720.0)))
		tmp = exp(-w);
	else
		tmp = l - ((l * w) + ((w * w) * (l * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 720.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l - N[(N[(l * w), $MachinePrecision] + N[(N[(w * w), $MachinePrecision] * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 720 < 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-/r/ 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. div-exp 100.0%

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

      2. associate-*r* 100.0%

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

      3. neg-mul-1 100.0%

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

      4. fma-neg 100.0%

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

      5. log-rec 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   if -0.680000000000000049 < w < 720

   1. Initial program 99.2%

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

      1. exp-neg 99.2%

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

      2. associate-*l/ 99.2%

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

      3. *-lft-identity 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in w around 0 93.9%

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

   5. Taylor expanded in w around 0 93.9%

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

      1. distribute-lft-out 93.9%

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

      2. unpow2 93.9%

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

      3. distribute-rgt-out 93.9%

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

      4. metadata-eval 93.9%

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

   7. Simplified 93.9%

      \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 720\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell - \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4: 97.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.5%

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

5. Final simplification 96.5%

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

Alternative 5: 73.7% accurate, 23.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.5%

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

5. Taylor expanded in w around 0 74.7%

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

   1. distribute-lft-out 74.7%

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

   2. unpow2 74.7%

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

   3. distribute-rgt-out 74.7%

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

   4. metadata-eval 74.7%

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

7. Simplified 74.7%

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

8. Final simplification 74.7%

   \[\leadsto \ell - \left(\ell \cdot w + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right) \]

Alternative 6: 63.6% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.024:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.024) (* l (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.024) {
		tmp = l * -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 <= (-0.024d0)) then
        tmp = l * -w
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.024) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.024:
		tmp = l * -w
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.024)
		tmp = Float64(l * Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.024)
		tmp = l * -w;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.024], N[(l * (-w)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.024

   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 0 100.0%

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

   5. Taylor expanded in w around 0 34.4%

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

      1. mul-1-neg 34.4%

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

      2. unsub-neg 34.4%

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

   7. Simplified 34.4%

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

   8. Taylor expanded in w around inf 34.4%

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

      1. associate-*r* 34.4%

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

      2. neg-mul-1 34.4%

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

   10. Simplified 34.4%

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

   if -0.024 < w

   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. associate-/r/ 99.1%

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

   3. Applied egg-rr 99.1%

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

   4. Taylor expanded in w around 0 76.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.024:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 7: 63.3% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.5%

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

5. Taylor expanded in w around 0 64.0%

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

   1. mul-1-neg 64.0%

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

   2. unsub-neg 64.0%

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

7. Simplified 64.0%

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

8. Taylor expanded in l around 0 64.0%

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

9. Final simplification 64.0%

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

Alternative 8: 63.3% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 96.5%

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

5. Taylor expanded in w around 0 64.0%

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

   1. mul-1-neg 64.0%

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

   2. unsub-neg 64.0%

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

7. Simplified 64.0%

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

8. Final simplification 64.0%

   \[\leadsto \ell - \ell \cdot w \]

Alternative 9: 56.3% 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 99.5%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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-/r/ 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in w around 0 55.6%

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

5. Final simplification 55.6%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023271 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))