exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 34.7s

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. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 97.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Taylor expanded in w around 0 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \ell \cdot e^{-w} \]
5. Add Preprocessing

Alternative 3: 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. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in w around 0 98.0%

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

6. Final simplification 98.0%

   \[\leadsto \frac{\ell}{e^{w}} \]
7. Add Preprocessing

Alternative 4: 89.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.145:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot \left(0.5 + w \cdot -1.1666666666666667\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.145)
   (+ l (* w (- (* l (* w (+ 0.5 (* w -1.1666666666666667)))) l)))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.145) {
		tmp = l + (w * ((l * (w * (0.5 + (w * -1.1666666666666667)))) - 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 <= 0.145d0) then
        tmp = l + (w * ((l * (w * (0.5d0 + (w * (-1.1666666666666667d0))))) - l))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.145) {
		tmp = l + (w * ((l * (w * (0.5 + (w * -1.1666666666666667)))) - l));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.145:
		tmp = l + (w * ((l * (w * (0.5 + (w * -1.1666666666666667)))) - l))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.145)
		tmp = Float64(l + Float64(w * Float64(Float64(l * Float64(w * Float64(0.5 + Float64(w * -1.1666666666666667)))) - l)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.145)
		tmp = l + (w * ((l * (w * (0.5 + (w * -1.1666666666666667)))) - l));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.145], N[(l + N[(w * N[(N[(l * N[(w * N[(0.5 + N[(w * -1.1666666666666667), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.14499999999999999

   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. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 31.7%

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

      2. sqrt-unprod 84.3%

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

      3. sqr-neg 84.3%

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

      4. sqrt-unprod 52.6%

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

      5. add-sqr-sqrt 83.5%

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

      6. add-sqr-sqrt 83.5%

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

      7. sqrt-unprod 83.5%

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

      8. exp-neg 83.5%

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

      9. inv-pow 83.5%

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

      10. add-sqr-sqrt 52.6%

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

      11. sqrt-unprod 83.5%

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

      12. sqr-neg 83.5%

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

      13. sqrt-unprod 30.9%

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

      14. add-sqr-sqrt 65.7%

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

      15. pow1 65.7%

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

      16. pow-prod-up 97.6%

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

      17. metadata-eval 97.6%

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

      18. metadata-eval 97.6%

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

      19. metadata-eval 97.6%

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

      20. pow1 97.6%

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

      21. add-exp-log 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in w around 0 85.6%

      \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
   8. Step-by-step derivation

      1. +-commutative 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(0.16666666666666666 \cdot \ell + -0.5 \cdot \ell\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      2. *-commutative 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(\color{blue}{\ell \cdot 0.16666666666666666} + -0.5 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      3. fma-define 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\mathsf{fma}\left(\ell, 0.16666666666666666, -0.5 \cdot \ell\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      4. metadata-eval 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(-1 \cdot 0.5\right)} \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      5. associate-*r* 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{-1 \cdot \left(0.5 \cdot \ell\right)}\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      6. *-commutative 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, -1 \cdot \color{blue}{\left(\ell \cdot 0.5\right)}\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      7. associate-*r* 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(-1 \cdot \ell\right) \cdot 0.5}\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}\right)} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      9. sqrt-unprod 70.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\sqrt{\left(-1 \cdot \ell\right) \cdot \left(-1 \cdot \ell\right)}} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      10. mul-1-neg 70.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \sqrt{\color{blue}{\left(-\ell\right)} \cdot \left(-1 \cdot \ell\right)} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      11. mul-1-neg 70.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \sqrt{\left(-\ell\right) \cdot \color{blue}{\left(-\ell\right)}} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      12. sqr-neg 70.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \sqrt{\color{blue}{\ell \cdot \ell}} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      13. sqrt-unprod 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      14. add-sqr-sqrt 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\ell} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. Applied egg-rr 85.6%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\mathsf{fma}\left(\ell, 0.16666666666666666, \ell \cdot 0.5\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
   10. Step-by-step derivation

       1. fma-undefine 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(\ell \cdot 0.16666666666666666 + \ell \cdot 0.5\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

       2. distribute-lft-out 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\ell \cdot \left(0.16666666666666666 + 0.5\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

       3. metadata-eval 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \ell \cdot \color{blue}{0.6666666666666666}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   11. Simplified 85.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\ell \cdot 0.6666666666666666}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   12. Taylor expanded in l around 0 86.4%

       \[\leadsto \ell + w \cdot \left(\color{blue}{\ell \cdot \left(w \cdot \left(0.5 + -1.1666666666666667 \cdot w\right)\right)} - \ell\right) \]

   if 0.14499999999999999 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 100.0%

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

   4. Taylor expanded in w around 0 5.3%

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

   6. Simplified 100.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.145:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot \left(0.5 + w \cdot -1.1666666666666667\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.102:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.102) (+ l (* w (- (* w (* l 0.5)) l))) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.102) {
		tmp = l + (w * ((w * (l * 0.5)) - 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 <= 0.102d0) then
        tmp = l + (w * ((w * (l * 0.5d0)) - l))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.102) {
		tmp = l + (w * ((w * (l * 0.5)) - l));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.102:
		tmp = l + (w * ((w * (l * 0.5)) - l))
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.102)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(l * 0.5)) - l)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.102)
		tmp = l + (w * ((w * (l * 0.5)) - l));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.102], N[(l + N[(w * N[(N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.101999999999999993

   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. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 31.7%

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

      2. sqrt-unprod 84.3%

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

      3. sqr-neg 84.3%

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

      4. sqrt-unprod 52.6%

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

      5. add-sqr-sqrt 83.5%

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

      6. add-sqr-sqrt 83.5%

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

      7. sqrt-unprod 83.5%

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

      8. exp-neg 83.5%

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

      9. inv-pow 83.5%

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

      10. add-sqr-sqrt 52.6%

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

      11. sqrt-unprod 83.5%

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

      12. sqr-neg 83.5%

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

      13. sqrt-unprod 30.9%

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

      14. add-sqr-sqrt 65.7%

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

      15. pow1 65.7%

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

      16. pow-prod-up 97.6%

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

      17. metadata-eval 97.6%

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

      18. metadata-eval 97.6%

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

      19. metadata-eval 97.6%

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

      20. pow1 97.6%

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

      21. add-exp-log 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in w around 0 85.6%

      \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]
   8. Step-by-step derivation

      1. +-commutative 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(0.16666666666666666 \cdot \ell + -0.5 \cdot \ell\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      2. *-commutative 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(\color{blue}{\ell \cdot 0.16666666666666666} + -0.5 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      3. fma-define 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\mathsf{fma}\left(\ell, 0.16666666666666666, -0.5 \cdot \ell\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      4. metadata-eval 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(-1 \cdot 0.5\right)} \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      5. associate-*r* 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{-1 \cdot \left(0.5 \cdot \ell\right)}\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      6. *-commutative 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, -1 \cdot \color{blue}{\left(\ell \cdot 0.5\right)}\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      7. associate-*r* 85.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(-1 \cdot \ell\right) \cdot 0.5}\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(\sqrt{-1 \cdot \ell} \cdot \sqrt{-1 \cdot \ell}\right)} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      9. sqrt-unprod 70.6%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\sqrt{\left(-1 \cdot \ell\right) \cdot \left(-1 \cdot \ell\right)}} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      10. mul-1-neg 70.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \sqrt{\color{blue}{\left(-\ell\right)} \cdot \left(-1 \cdot \ell\right)} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      11. mul-1-neg 70.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \sqrt{\left(-\ell\right) \cdot \color{blue}{\left(-\ell\right)}} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      12. sqr-neg 70.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \sqrt{\color{blue}{\ell \cdot \ell}} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      13. sqrt-unprod 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

      14. add-sqr-sqrt 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \mathsf{fma}\left(\ell, 0.16666666666666666, \color{blue}{\ell} \cdot 0.5\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. Applied egg-rr 85.6%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\mathsf{fma}\left(\ell, 0.16666666666666666, \ell \cdot 0.5\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
   10. Step-by-step derivation

       1. fma-undefine 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\left(\ell \cdot 0.16666666666666666 + \ell \cdot 0.5\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

       2. distribute-lft-out 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\ell \cdot \left(0.16666666666666666 + 0.5\right)}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

       3. metadata-eval 85.6%

          \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \ell \cdot \color{blue}{0.6666666666666666}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   11. Simplified 85.6%

       \[\leadsto \ell + w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \color{blue}{\ell \cdot 0.6666666666666666}\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   12. Taylor expanded in w around 0 81.5%

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

       1. mul-1-neg 81.5%

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

       2. distribute-rgt-neg-in 81.5%

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

       3. distribute-rgt-out 81.5%

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

       4. metadata-eval 81.5%

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

       5. distribute-rgt-neg-in 81.5%

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

       6. metadata-eval 81.5%

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

   14. Simplified 81.5%

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

   if 0.101999999999999993 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 100.0%

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

   4. Taylor expanded in w around 0 5.3%

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

   6. Simplified 100.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.102:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.061:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{elif}\;w \leq 0.088:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.061) (* l (- w)) (if (<= w 0.088) l 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.061) {
		tmp = l * -w;
	} else if (w <= 0.088) {
		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 <= (-0.061d0)) then
        tmp = l * -w
    else if (w <= 0.088d0) 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 <= -0.061) {
		tmp = l * -w;
	} else if (w <= 0.088) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.061:
		tmp = l * -w
	elif w <= 0.088:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.061)
		tmp = Float64(l * Float64(-w));
	elseif (w <= 0.088)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.061)
		tmp = l * -w;
	elseif (w <= 0.088)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.061], N[(l * (-w)), $MachinePrecision], If[LessEqual[w, 0.088], l, 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.060999999999999999

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 100.0%

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

   4. Taylor expanded in w around 0 25.5%

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

   5. Taylor expanded in w around inf 25.5%

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

      1. associate-*r* 25.5%

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

      2. neg-mul-1 25.5%

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

   7. Simplified 25.5%

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

   if -0.060999999999999999 < w < 0.087999999999999995

   1. Initial program 99.2%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 96.5%

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

   4. Taylor expanded in w around 0 96.5%

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

   if 0.087999999999999995 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 100.0%

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

   4. Taylor expanded in w around 0 5.3%

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

   6. Simplified 100.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.061:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{elif}\;w \leq 0.088:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.17) (* l (- 1.0 w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l * (1.0 - 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 <= 0.17d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.17) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.17:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.17)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.17)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.17], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.170000000000000012

   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. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 97.6%

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

   6. Taylor expanded in w around 0 73.9%

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

      1. mul-1-neg 73.9%

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

      2. sub-neg 73.9%

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

      3. *-rgt-identity 73.9%

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

      4. distribute-lft-out-- 73.9%

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

   8. Simplified 73.9%

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

   if 0.170000000000000012 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 100.0%

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

   4. Taylor expanded in w around 0 5.3%

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

   6. Simplified 100.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.17:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.0% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.12) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.12) {
		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 <= 0.12d0) 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 <= 0.12) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.12:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.12)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.12)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.12], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.12

   1. Initial program 99.5%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 97.6%

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

   4. Taylor expanded in w around 0 67.0%

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

   if 0.12 < w

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 100.0%

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

   4. Taylor expanded in w around 0 5.3%

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

   6. Simplified 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.12:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 16.9% 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 99.5%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Taylor expanded in w around 0 98.0%

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

4. Taylor expanded in w around 0 56.7%

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

6. Simplified 19.5%

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

7. Final simplification 19.5%

   \[\leadsto 0 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))