exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 17.4s

Alternatives: 10

Speedup: N/A×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% 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 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 97.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 440000000:\\ \;\;\;\;\ell \cdot \left(1 - \left(w + w \cdot \left(w \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (exp (- w))
   (if (<= w 440000000.0) (* l (- 1.0 (+ w (* w (* w -0.5))))) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp(-w);
	} else if (w <= 440000000.0) {
		tmp = l * (1.0 - (w + (w * (w * -0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.7d0)) then
        tmp = exp(-w)
    else if (w <= 440000000.0d0) then
        tmp = l * (1.0d0 - (w + (w * (w * (-0.5d0)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = Math.exp(-w);
	} else if (w <= 440000000.0) {
		tmp = l * (1.0 - (w + (w * (w * -0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.7:
		tmp = math.exp(-w)
	elif w <= 440000000.0:
		tmp = l * (1.0 - (w + (w * (w * -0.5))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(-w));
	elseif (w <= 440000000.0)
		tmp = Float64(l * Float64(1.0 - Float64(w + Float64(w * Float64(w * -0.5)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.7)
		tmp = exp(-w);
	elseif (w <= 440000000.0)
		tmp = l * (1.0 - (w + (w * (w * -0.5))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 440000000.0], N[(l * N[(1.0 - N[(w + N[(w * N[(w * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   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. remove-double-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. flip-- 0.0%

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

      4. sqr-neg 0.0%

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

      5. remove-double-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. sub-neg 0.0%

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

      8. flip-+ 100.0%

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

      9. +-commutative 100.0%

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

      10. /-rgt-identity 100.0%

          \[\leadsto \frac{{\ell}^{\left(\color{blue}{\frac{e^{w}}{1}} + 0\right)}}{e^{w}} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \color{blue}{\left(0 - 0\right)}\right)}}{e^{w}} \]

      12. flip3-- 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \color{blue}{\frac{{0}^{3} - {0}^{3}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}}\right)}}{e^{w}} \]

      13. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{\color{blue}{0} - {0}^{3}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      14. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0 - \color{blue}{0}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      15. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{\color{blue}{0}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      16. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{\color{blue}{0} + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      17. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{0 + \left(\color{blue}{0} + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      18. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{0 + \left(0 + \color{blue}{0}\right)}\right)}}{e^{w}} \]

      19. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{0 + \color{blue}{0}}\right)}}{e^{w}} \]

      20. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{\color{blue}{0}}\right)}}{e^{w}} \]

   5. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. rec-exp 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. *-lft-identity 100.0%

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

      3. exp-neg 100.0%

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

   9. Simplified 100.0%

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

   if -0.69999999999999996 < w < 4.4e8

   1. Initial program 96.9%

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

      1. exp-neg 96.9%

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

      2. associate-*l/ 96.9%

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

      3. *-lft-identity 96.9%

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

   3. Simplified 96.9%

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

      1. add-cube-cbrt_binary64 94.9%

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

   5. Applied rewrite-once 94.9%

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

   6. Taylor expanded in w around 0 94.4%

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

   7. Taylor expanded in w around 0 94.5%

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

      1. distribute-lft-out 94.5%

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

      2. unpow2 94.5%

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

      3. distribute-rgt-out 94.5%

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

      4. metadata-eval 94.5%

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

   9. Simplified 94.5%

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

   10. Taylor expanded in l around 0 94.5%

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

       1. mul-1-neg 94.5%

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

       2. unsub-neg 94.5%

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

       3. *-commutative 94.5%

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

       4. unpow2 94.5%

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

       5. associate-*r* 94.5%

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

   12. Simplified 94.5%

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

   if 4.4e8 < w

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 440000000:\\ \;\;\;\;\ell \cdot \left(1 - \left(w + w \cdot \left(w \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 97.4% 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 98.5%

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

   1. exp-neg 98.5%

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

   2. associate-*l/ 98.5%

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

   3. *-lft-identity 98.5%

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

3. Simplified 98.5%

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

   1. add-cube-cbrt_binary64 97.5%

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

5. Applied rewrite-once 97.5%

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

6. Taylor expanded in w around 0 97.2%

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

7. Final simplification 97.2%

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

Alternative 4: 87.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 440000000:\\ \;\;\;\;\ell \cdot \left(1 - \left(w + w \cdot \left(w \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 440000000.0) (* l (- 1.0 (+ w (* w (* w -0.5))))) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 440000000.0) {
		tmp = l * (1.0 - (w + (w * (w * -0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 440000000.0d0) then
        tmp = l * (1.0d0 - (w + (w * (w * (-0.5d0)))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 440000000.0) {
		tmp = l * (1.0 - (w + (w * (w * -0.5))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 440000000.0:
		tmp = l * (1.0 - (w + (w * (w * -0.5))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 440000000.0)
		tmp = Float64(l * Float64(1.0 - Float64(w + Float64(w * Float64(w * -0.5)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 440000000.0)
		tmp = l * (1.0 - (w + (w * (w * -0.5))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 440000000.0], N[(l * N[(1.0 - N[(w + N[(w * N[(w * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 4.4e8

   1. Initial program 98.1%

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

      1. exp-neg 98.1%

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

      2. associate-*l/ 98.1%

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

      3. *-lft-identity 98.1%

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

   3. Simplified 98.1%

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

      1. add-cube-cbrt_binary64 96.9%

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

   5. Applied rewrite-once 96.9%

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

   6. Taylor expanded in w around 0 96.6%

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

   7. Taylor expanded in w around 0 83.0%

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

      1. distribute-lft-out 83.0%

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

      2. unpow2 83.0%

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

      3. distribute-rgt-out 83.0%

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

      4. metadata-eval 83.0%

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

   9. Simplified 83.0%

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

   10. Taylor expanded in l around 0 83.0%

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

       1. mul-1-neg 83.0%

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

       2. unsub-neg 83.0%

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

       3. *-commutative 83.0%

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

       4. unpow2 83.0%

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

       5. associate-*r* 83.0%

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

   12. Simplified 83.0%

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

   if 4.4e8 < w

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 440000000:\\ \;\;\;\;\ell \cdot \left(1 - \left(w + w \cdot \left(w \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 87.4% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.74:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right) - w\right)\\ \mathbf{elif}\;w \leq 440000000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.74) (* l (- (* w (* w 0.5)) w)) (if (<= w 440000000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.74) {
		tmp = l * ((w * (w * 0.5)) - w);
	} else if (w <= 440000000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.74d0)) then
        tmp = l * ((w * (w * 0.5d0)) - w)
    else if (w <= 440000000.0d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.74) {
		tmp = l * ((w * (w * 0.5)) - w);
	} else if (w <= 440000000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.74:
		tmp = l * ((w * (w * 0.5)) - w)
	elif w <= 440000000.0:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.74)
		tmp = Float64(l * Float64(Float64(w * Float64(w * 0.5)) - w));
	elseif (w <= 440000000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.74)
		tmp = l * ((w * (w * 0.5)) - w);
	elseif (w <= 440000000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.74], N[(l * N[(N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 440000000.0], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.73999999999999999

   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-cube-cbrt_binary64 100.0%

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

   5. Applied rewrite-once 100.0%

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

   6. Taylor expanded in w around 0 100.0%

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

   7. Taylor expanded in w around 0 65.0%

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

      1. distribute-lft-out 65.0%

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

      2. unpow2 65.0%

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

      3. distribute-rgt-out 65.0%

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

      4. metadata-eval 65.0%

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

   9. Simplified 65.0%

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

   10. Taylor expanded in w around inf 65.0%

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

       1. *-commutative 65.0%

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

       2. unpow2 65.0%

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

       3. associate-*r* 65.0%

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

       4. mul-1-neg 65.0%

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

       5. distribute-rgt-neg-out 65.0%

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

       6. unpow2 65.0%

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

       7. *-commutative 65.0%

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

       8. distribute-lft-in 65.0%

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

       9. +-commutative 65.0%

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

       10. unsub-neg 65.0%

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

       11. *-commutative 65.0%

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

       12. unpow2 65.0%

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

       13. associate-*l* 65.0%

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

   12. Simplified 65.0%

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

   if -0.73999999999999999 < w < 4.4e8

   1. Initial program 96.9%

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

   2. Taylor expanded in w around 0 94.5%

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

   if 4.4e8 < w

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.74:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right) - w\right)\\ \mathbf{elif}\;w \leq 440000000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 87.4% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.5:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right)\right)\\ \mathbf{elif}\;w \leq 440000000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.5) (* l (* w (* w 0.5))) (if (<= w 440000000.0) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.5) {
		tmp = l * (w * (w * 0.5));
	} else if (w <= 440000000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.5d0)) then
        tmp = l * (w * (w * 0.5d0))
    else if (w <= 440000000.0d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.5) {
		tmp = l * (w * (w * 0.5));
	} else if (w <= 440000000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.5:
		tmp = l * (w * (w * 0.5))
	elif w <= 440000000.0:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.5)
		tmp = Float64(l * Float64(w * Float64(w * 0.5)));
	elseif (w <= 440000000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.5)
		tmp = l * (w * (w * 0.5));
	elseif (w <= 440000000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.5], N[(l * N[(w * N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 440000000.0], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.5

   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-cube-cbrt_binary64 100.0%

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

   5. Applied rewrite-once 100.0%

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

   6. Taylor expanded in w around 0 100.0%

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

   7. Taylor expanded in w around 0 65.0%

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

      1. distribute-lft-out 65.0%

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

      2. unpow2 65.0%

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

      3. distribute-rgt-out 65.0%

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

      4. metadata-eval 65.0%

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

   9. Simplified 65.0%

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

   10. Taylor expanded in w around inf 65.0%

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

       1. *-commutative 65.0%

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

       2. unpow2 65.0%

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

       3. associate-*r* 65.0%

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

       4. associate-*l* 65.0%

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

   12. Simplified 65.0%

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

   if -0.5 < w < 4.4e8

   1. Initial program 96.9%

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

   2. Taylor expanded in w around 0 94.5%

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

   if 4.4e8 < w

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.5:\\ \;\;\;\;\ell \cdot \left(w \cdot \left(w \cdot 0.5\right)\right)\\ \mathbf{elif}\;w \leq 440000000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 77.5% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.215:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.215) (- l (* l w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.215) {
		tmp = l - (l * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.215d0) then
        tmp = l - (l * w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.215) {
		tmp = l - (l * w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.215:
		tmp = l - (l * w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.215)
		tmp = Float64(l - Float64(l * w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.215)
		tmp = l - (l * w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.215], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.214999999999999997

   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-cube-cbrt_binary64 98.3%

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

   5. Applied rewrite-once 98.3%

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

   6. Taylor expanded in w around 0 98.0%

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

   7. Taylor expanded in w around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. unsub-neg 72.5%

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

   9. Simplified 72.5%

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

   if 0.214999999999999997 < w

   1. Initial program 93.9%

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

   3. Applied egg-rr 94.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.215:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 17.9% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 440000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 440000000.0) 1.0 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 440000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 440000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 440000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 440000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 440000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 440000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 440000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 4.4e8

   1. Initial program 98.1%

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

      1. exp-neg 98.1%

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

      2. associate-*l/ 98.1%

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

      3. *-lft-identity 98.1%

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

   3. Simplified 98.1%

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

      1. remove-double-neg 98.1%

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

      2. neg-sub0 98.1%

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

      3. flip-- 59.1%

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

      4. sqr-neg 59.1%

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

      5. remove-double-neg 59.1%

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

      6. remove-double-neg 59.1%

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

      7. sub-neg 59.1%

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

      8. flip-+ 98.1%

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

      9. +-commutative 98.1%

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

      10. /-rgt-identity 98.1%

          \[\leadsto \frac{{\ell}^{\left(\color{blue}{\frac{e^{w}}{1}} + 0\right)}}{e^{w}} \]

      11. metadata-eval 98.1%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \color{blue}{\left(0 - 0\right)}\right)}}{e^{w}} \]

      12. flip3-- 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \color{blue}{\frac{{0}^{3} - {0}^{3}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}}\right)}}{e^{w}} \]

      13. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{\color{blue}{0} - {0}^{3}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      14. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0 - \color{blue}{0}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      15. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{\color{blue}{0}}{0 \cdot 0 + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      16. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{\color{blue}{0} + \left(0 \cdot 0 + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      17. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{0 + \left(\color{blue}{0} + 0 \cdot 0\right)}\right)}}{e^{w}} \]

      18. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{0 + \left(0 + \color{blue}{0}\right)}\right)}}{e^{w}} \]

      19. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{0 + \color{blue}{0}}\right)}}{e^{w}} \]

      20. metadata-eval 0.0%

          \[\leadsto \frac{{\ell}^{\left(\frac{e^{w}}{1} + \frac{0}{\color{blue}{0}}\right)}}{e^{w}} \]

   5. Applied egg-rr 42.6%

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

   6. Taylor expanded in w around 0 4.8%

      \[\leadsto \color{blue}{1} \]

   if 4.4e8 < w

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 440000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 70.2% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 440000000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 440000000.0) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 440000000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 440000000.0d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 440000000.0) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 440000000.0:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 440000000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 440000000.0)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 440000000.0], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 4.4e8

   1. Initial program 98.1%

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

   2. Taylor expanded in w around 0 59.1%

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

   if 4.4e8 < w

   1. Initial program 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 440000000:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 17.0% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 0.0)

C

double code(double w, double l) {
	return 0.0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 0.0d0
end function

Java

public static double code(double w, double l) {
	return 0.0;
}

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

function tmp = code(w, l)
	tmp = 0.0;
end

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 98.5%

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

3. Applied egg-rr 20.3%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 20.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))