exp-w (used to crash)

Percentage Accurate: 99.4% → 99.1%

Time: 23.9s

Alternatives: 13

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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\ell \cdot {\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{\left(e^{w} + 1\right) + -1} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (pow
   (* l (pow (pow l w) (fma w (fma w 0.020833333333333332 0.125) 0.5)))
   (sqrt (exp w)))
  (+ (+ (exp w) 1.0) -1.0)))

C

double code(double w, double l) {
	return pow((l * pow(pow(l, w), fma(w, fma(w, 0.020833333333333332, 0.125), 0.5))), sqrt(exp(w))) / ((exp(w) + 1.0) + -1.0);
}

Julia

function code(w, l)
	return Float64((Float64(l * ((l ^ w) ^ fma(w, fma(w, 0.020833333333333332, 0.125), 0.5))) ^ sqrt(exp(w))) / Float64(Float64(exp(w) + 1.0) + -1.0))
end

Wolfram

code[w_, l_] := N[(N[Power[N[(l * N[Power[N[Power[l, w], $MachinePrecision], N[(w * N[(w * 0.020833333333333332 + 0.125), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[(N[Exp[w], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow-unpow 99.8%

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

   3. pow-to-exp 94.9%

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

6. Applied egg-rr 94.9%

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

   1. exp-to-pow 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in w around 0 99.8%

   \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + 0.020833333333333332 \cdot w\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
10. Step-by-step derivation

    1. *-commutative 99.8%

       \[\leadsto \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + \color{blue}{w \cdot 0.020833333333333332}\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

11. Simplified 99.8%

    \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
12. Step-by-step derivation

    1. +-commutative 99.8%

       \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right) + 1\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    2. unpow-prod-up 99.9%

       \[\leadsto \frac{{\color{blue}{\left({\ell}^{\left(w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)} \cdot {\ell}^{1}\right)}}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    3. pow-unpow 99.9%

       \[\leadsto \frac{{\left(\color{blue}{{\left({\ell}^{w}\right)}^{\left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)}} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    4. +-commutative 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\color{blue}{\left(w \cdot \left(0.125 + w \cdot 0.020833333333333332\right) + 0.5\right)}} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    5. fma-define 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\color{blue}{\left(\mathsf{fma}\left(w, 0.125 + w \cdot 0.020833333333333332, 0.5\right)\right)}} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    6. +-commutative 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \color{blue}{w \cdot 0.020833333333333332 + 0.125}, 0.5\right)\right)} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    7. fma-define 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \color{blue}{\mathsf{fma}\left(w, 0.020833333333333332, 0.125\right)}, 0.5\right)\right)} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    8. pow1 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \color{blue}{\ell}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

13. Applied egg-rr 99.9%

    \[\leadsto \frac{{\color{blue}{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
14. Step-by-step derivation

    1. expm1-log1p-u 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{w}\right)\right)}} \]

    2. expm1-undefine 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{e^{\mathsf{log1p}\left(e^{w}\right)} - 1}} \]

    3. log1p-undefine 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{\color{blue}{\log \left(1 + e^{w}\right)}} - 1} \]

    4. rem-exp-log 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\left(1 + e^{w}\right)} - 1} \]

15. Applied egg-rr 99.9%

    \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\left(1 + e^{w}\right) - 1}} \]

16. Final simplification 99.9%

    \[\leadsto \frac{{\left(\ell \cdot {\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{\left(e^{w} + 1\right) + -1} \]
17. Add Preprocessing

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\ell \cdot {\left({\ell}^{w}\right)}^{\left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (pow
   (* l (pow (pow l w) (+ 0.5 (* w (+ 0.125 (* w 0.020833333333333332))))))
   (sqrt (exp w)))
  (exp w)))

C

double code(double w, double l) {
	return pow((l * pow(pow(l, w), (0.5 + (w * (0.125 + (w * 0.020833333333333332)))))), sqrt(exp(w))) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l * ((l ** w) ** (0.5d0 + (w * (0.125d0 + (w * 0.020833333333333332d0)))))) ** sqrt(exp(w))) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow((l * Math.pow(Math.pow(l, w), (0.5 + (w * (0.125 + (w * 0.020833333333333332)))))), Math.sqrt(Math.exp(w))) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow((l * math.pow(math.pow(l, w), (0.5 + (w * (0.125 + (w * 0.020833333333333332)))))), math.sqrt(math.exp(w))) / math.exp(w)

Julia

function code(w, l)
	return Float64((Float64(l * ((l ^ w) ^ Float64(0.5 + Float64(w * Float64(0.125 + Float64(w * 0.020833333333333332)))))) ^ sqrt(exp(w))) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l * ((l ^ w) ^ (0.5 + (w * (0.125 + (w * 0.020833333333333332)))))) ^ sqrt(exp(w))) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[N[(l * N[Power[N[Power[l, w], $MachinePrecision], N[(0.5 + N[(w * N[(0.125 + N[(w * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow-unpow 99.8%

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

   3. pow-to-exp 94.9%

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

6. Applied egg-rr 94.9%

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

   1. exp-to-pow 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in w around 0 99.8%

   \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + 0.020833333333333332 \cdot w\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
10. Step-by-step derivation

    1. *-commutative 99.8%

       \[\leadsto \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + \color{blue}{w \cdot 0.020833333333333332}\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

11. Simplified 99.8%

    \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
12. Step-by-step derivation

    1. +-commutative 99.8%

       \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right) + 1\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    2. unpow-prod-up 99.9%

       \[\leadsto \frac{{\color{blue}{\left({\ell}^{\left(w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)} \cdot {\ell}^{1}\right)}}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    3. pow-unpow 99.9%

       \[\leadsto \frac{{\left(\color{blue}{{\left({\ell}^{w}\right)}^{\left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)}} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    4. +-commutative 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\color{blue}{\left(w \cdot \left(0.125 + w \cdot 0.020833333333333332\right) + 0.5\right)}} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    5. fma-define 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\color{blue}{\left(\mathsf{fma}\left(w, 0.125 + w \cdot 0.020833333333333332, 0.5\right)\right)}} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    6. +-commutative 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \color{blue}{w \cdot 0.020833333333333332 + 0.125}, 0.5\right)\right)} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    7. fma-define 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \color{blue}{\mathsf{fma}\left(w, 0.020833333333333332, 0.125\right)}, 0.5\right)\right)} \cdot {\ell}^{1}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

    8. pow1 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \color{blue}{\ell}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

13. Applied egg-rr 99.9%

    \[\leadsto \frac{{\color{blue}{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

14. Taylor expanded in w around 0 99.9%

    \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\color{blue}{\left(0.5 + w \cdot \left(0.125 + 0.020833333333333332 \cdot w\right)\right)}} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

15. Final simplification 99.9%

    \[\leadsto \frac{{\left(\ell \cdot {\left({\ell}^{w}\right)}^{\left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
16. Add Preprocessing

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{\left(e^{w} + 1\right) + -1} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (pow
   (pow l (+ 1.0 (* w (+ 0.5 (* w (+ 0.125 (* w 0.020833333333333332)))))))
   (sqrt (exp w)))
  (+ (+ (exp w) 1.0) -1.0)))

C

double code(double w, double l) {
	return pow(pow(l, (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))), sqrt(exp(w))) / ((exp(w) + 1.0) + -1.0);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l ** (1.0d0 + (w * (0.5d0 + (w * (0.125d0 + (w * 0.020833333333333332d0))))))) ** sqrt(exp(w))) / ((exp(w) + 1.0d0) + (-1.0d0))
end function

Java

public static double code(double w, double l) {
	return Math.pow(Math.pow(l, (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))), Math.sqrt(Math.exp(w))) / ((Math.exp(w) + 1.0) + -1.0);
}

Python

def code(w, l):
	return math.pow(math.pow(l, (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))), math.sqrt(math.exp(w))) / ((math.exp(w) + 1.0) + -1.0)

Julia

function code(w, l)
	return Float64(((l ^ Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * Float64(0.125 + Float64(w * 0.020833333333333332))))))) ^ sqrt(exp(w))) / Float64(Float64(exp(w) + 1.0) + -1.0))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l ^ (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))) ^ sqrt(exp(w))) / ((exp(w) + 1.0) + -1.0);
end

Wolfram

code[w_, l_] := N[(N[Power[N[Power[l, N[(1.0 + N[(w * N[(0.5 + N[(w * N[(0.125 + N[(w * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[(N[Exp[w], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow-unpow 99.8%

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

   3. pow-to-exp 94.9%

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

6. Applied egg-rr 94.9%

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

   1. exp-to-pow 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in w around 0 99.8%

   \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + 0.020833333333333332 \cdot w\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
10. Step-by-step derivation

    1. *-commutative 99.8%

       \[\leadsto \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + \color{blue}{w \cdot 0.020833333333333332}\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

11. Simplified 99.8%

    \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
12. Step-by-step derivation

    1. expm1-log1p-u 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{w}\right)\right)}} \]

    2. expm1-undefine 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{e^{\mathsf{log1p}\left(e^{w}\right)} - 1}} \]

    3. log1p-undefine 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{\color{blue}{\log \left(1 + e^{w}\right)}} - 1} \]

    4. rem-exp-log 99.9%

       \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\left(1 + e^{w}\right)} - 1} \]

13. Applied egg-rr 99.8%

    \[\leadsto \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\left(1 + e^{w}\right) - 1}} \]

14. Final simplification 99.8%

    \[\leadsto \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{\left(e^{w} + 1\right) + -1} \]
15. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (pow
   (pow l (+ 1.0 (* w (+ 0.5 (* w (+ 0.125 (* w 0.020833333333333332)))))))
   (sqrt (exp w)))
  (exp w)))

C

double code(double w, double l) {
	return pow(pow(l, (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))), sqrt(exp(w))) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l ** (1.0d0 + (w * (0.5d0 + (w * (0.125d0 + (w * 0.020833333333333332d0))))))) ** sqrt(exp(w))) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow(Math.pow(l, (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))), Math.sqrt(Math.exp(w))) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(math.pow(l, (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))), math.sqrt(math.exp(w))) / math.exp(w)

Julia

function code(w, l)
	return Float64(((l ^ Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * Float64(0.125 + Float64(w * 0.020833333333333332))))))) ^ sqrt(exp(w))) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l ^ (1.0 + (w * (0.5 + (w * (0.125 + (w * 0.020833333333333332))))))) ^ sqrt(exp(w))) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[N[Power[l, N[(1.0 + N[(w * N[(0.5 + N[(w * N[(0.125 + N[(w * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow-unpow 99.8%

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

   3. pow-to-exp 94.9%

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

6. Applied egg-rr 94.9%

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

   1. exp-to-pow 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in w around 0 99.8%

   \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + 0.020833333333333332 \cdot w\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
10. Step-by-step derivation

    1. *-commutative 99.8%

       \[\leadsto \frac{{\left({\ell}^{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + \color{blue}{w \cdot 0.020833333333333332}\right)\right)\right)}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]

11. Simplified 99.8%

    \[\leadsto \frac{{\left({\ell}^{\color{blue}{\left(1 + w \cdot \left(0.5 + w \cdot \left(0.125 + w \cdot 0.020833333333333332\right)\right)\right)}}\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{w}} \]
12. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{w} + 1\right) + -1} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (+ (+ (exp w) 1.0) -1.0)))

C

double code(double w, double l) {
	return pow(l, exp(w)) / ((exp(w) + 1.0) + -1.0);
}

Fortran

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

Java

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

Python

def code(w, l):
	return math.pow(l, math.exp(w)) / ((math.exp(w) + 1.0) + -1.0)

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / Float64(Float64(exp(w) + 1.0) + -1.0))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / ((exp(w) + 1.0) + -1.0);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 99.9%

      \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{w}\right)\right)}} \]

   2. expm1-undefine 99.9%

      \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{e^{\mathsf{log1p}\left(e^{w}\right)} - 1}} \]

   3. log1p-undefine 99.9%

      \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{e^{\color{blue}{\log \left(1 + e^{w}\right)}} - 1} \]

   4. rem-exp-log 99.9%

      \[\leadsto \frac{{\left({\left({\ell}^{w}\right)}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.020833333333333332, 0.125\right), 0.5\right)\right)} \cdot \ell\right)}^{\left(\sqrt{e^{w}}\right)}}{\color{blue}{\left(1 + e^{w}\right)} - 1} \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{w} + 1\right) + -1} \]
8. Add Preprocessing

Alternative 6: 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 7: 97.9% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 99.0%

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

6. Taylor expanded in l around 0 99.0%

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

Alternative 8: 88.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot 0.5 - \ell\\ \mathbf{if}\;w \leq 0.106:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(t\_0 - w \cdot \left(t\_0 - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(-1 + \left(1 - \ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- (* l 0.5) l)))
   (if (<= w 0.106)
     (-
      l
      (*
       w
       (+
        l
        (* w (- t_0 (* w (- t_0 (+ (* l -0.5) (* l 0.16666666666666666)))))))))
     (* w (+ -1.0 (- 1.0 l))))))

C

double code(double w, double l) {
	double t_0 = (l * 0.5) - l;
	double tmp;
	if (w <= 0.106) {
		tmp = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))));
	} else {
		tmp = w * (-1.0 + (1.0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * 0.5d0) - l
    if (w <= 0.106d0) then
        tmp = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * (-0.5d0)) + (l * 0.16666666666666666d0))))))))
    else
        tmp = w * ((-1.0d0) + (1.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = (l * 0.5) - l;
	double tmp;
	if (w <= 0.106) {
		tmp = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))));
	} else {
		tmp = w * (-1.0 + (1.0 - l));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = (l * 0.5) - l
	tmp = 0
	if w <= 0.106:
		tmp = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))))
	else:
		tmp = w * (-1.0 + (1.0 - l))
	return tmp

Julia

function code(w, l)
	t_0 = Float64(Float64(l * 0.5) - l)
	tmp = 0.0
	if (w <= 0.106)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(t_0 - Float64(w * Float64(t_0 - Float64(Float64(l * -0.5) + Float64(l * 0.16666666666666666)))))))));
	else
		tmp = Float64(w * Float64(-1.0 + Float64(1.0 - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = (l * 0.5) - l;
	tmp = 0.0;
	if (w <= 0.106)
		tmp = l - (w * (l + (w * (t_0 - (w * (t_0 - ((l * -0.5) + (l * 0.16666666666666666))))))));
	else
		tmp = w * (-1.0 + (1.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[w, 0.106], N[(l - N[(w * N[(l + N[(w * N[(t$95$0 - N[(w * N[(t$95$0 - N[(N[(l * -0.5), $MachinePrecision] + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(-1.0 + N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 0.105999999999999997

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 98.8%

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

   6. Taylor expanded in w around 0 89.6%

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

   if 0.105999999999999997 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. unsub-neg 3.4%

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

      3. *-commutative 3.4%

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

   8. Simplified 3.4%

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

   9. Taylor expanded in w around inf 3.4%

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

       1. neg-mul-1 3.4%

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

       2. distribute-lft-neg-in 3.4%

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

   11. Simplified 3.4%

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

       1. expm1-log1p-u 3.4%

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

       2. expm1-undefine 97.6%

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

   13. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \cdot w \]
   14. Step-by-step derivation

       1. sub-neg 97.6%

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

       2. metadata-eval 97.6%

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

       3. +-commutative 97.6%

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

       4. log1p-undefine 97.6%

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

       5. rem-exp-log 97.6%

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

       6. unsub-neg 97.6%

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

   15. Simplified 97.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.106:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) - w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(-1 + \left(1 - \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.059:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(-1 + \left(1 - \ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.059) (- l (* w (+ l (* w (* l -0.5))))) (* w (+ -1.0 (- 1.0 l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.059) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = w * (-1.0 + (1.0 - 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 <= 0.059d0) then
        tmp = l - (w * (l + (w * (l * (-0.5d0)))))
    else
        tmp = w * ((-1.0d0) + (1.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.059) {
		tmp = l - (w * (l + (w * (l * -0.5))));
	} else {
		tmp = w * (-1.0 + (1.0 - l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.059:
		tmp = l - (w * (l + (w * (l * -0.5))))
	else:
		tmp = w * (-1.0 + (1.0 - l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.059)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(l * -0.5)))));
	else
		tmp = Float64(w * Float64(-1.0 + Float64(1.0 - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.059)
		tmp = l - (w * (l + (w * (l * -0.5))));
	else
		tmp = w * (-1.0 + (1.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.059], N[(l - N[(w * N[(l + N[(w * N[(l * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(-1.0 + N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.058999999999999997

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 98.8%

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

   6. Taylor expanded in w around 0 83.8%

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

      1. associate-*r* 83.8%

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

      2. neg-mul-1 83.8%

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

      3. distribute-rgt-out 83.8%

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

      4. metadata-eval 83.8%

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

   8. Simplified 83.8%

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

   if 0.058999999999999997 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. unsub-neg 3.4%

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

      3. *-commutative 3.4%

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

   8. Simplified 3.4%

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

   9. Taylor expanded in w around inf 3.4%

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

       1. neg-mul-1 3.4%

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

       2. distribute-lft-neg-in 3.4%

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

   11. Simplified 3.4%

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

       1. expm1-log1p-u 3.4%

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

       2. expm1-undefine 97.6%

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

   13. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \cdot w \]
   14. Step-by-step derivation

       1. sub-neg 97.6%

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

       2. metadata-eval 97.6%

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

       3. +-commutative 97.6%

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

       4. log1p-undefine 97.6%

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

       5. rem-exp-log 97.6%

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

       6. unsub-neg 97.6%

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

   15. Simplified 97.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.059:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\ell \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(-1 + \left(1 - \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.065:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(-1 + \left(1 - \ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.065) (- l (* l w)) (* w (+ -1.0 (- 1.0 l)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.065) {
		tmp = l - (l * w);
	} else {
		tmp = w * (-1.0 + (1.0 - l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.065d0) then
        tmp = l - (l * w)
    else
        tmp = w * ((-1.0d0) + (1.0d0 - l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.065) {
		tmp = l - (l * w);
	} else {
		tmp = w * (-1.0 + (1.0 - l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.065:
		tmp = l - (l * w)
	else:
		tmp = w * (-1.0 + (1.0 - l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.065)
		tmp = Float64(l - Float64(l * w));
	else
		tmp = Float64(w * Float64(-1.0 + Float64(1.0 - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.065)
		tmp = l - (l * w);
	else
		tmp = w * (-1.0 + (1.0 - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.065], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision], N[(w * N[(-1.0 + N[(1.0 - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.065000000000000002

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 98.8%

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

   6. Taylor expanded in w around 0 73.7%

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

      1. mul-1-neg 73.7%

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

      2. unsub-neg 73.7%

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

      3. *-commutative 73.7%

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

   8. Simplified 73.7%

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

   if 0.065000000000000002 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. unsub-neg 3.4%

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

      3. *-commutative 3.4%

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

   8. Simplified 3.4%

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

   9. Taylor expanded in w around inf 3.4%

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

       1. neg-mul-1 3.4%

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

       2. distribute-lft-neg-in 3.4%

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

   11. Simplified 3.4%

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

       1. expm1-log1p-u 3.4%

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

       2. expm1-undefine 97.6%

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

   13. Applied egg-rr 97.6%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(-\ell\right)} - 1\right)} \cdot w \]
   14. Step-by-step derivation

       1. sub-neg 97.6%

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

       2. metadata-eval 97.6%

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

       3. +-commutative 97.6%

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

       4. log1p-undefine 97.6%

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

       5. rem-exp-log 97.6%

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

       6. unsub-neg 97.6%

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

   15. Simplified 97.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.065:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(-1 + \left(1 - \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.3% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.41:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.41) (* l (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.41) {
		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 <= (-0.41d0)) 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 <= -0.41) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.41:
		tmp = l * -w
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.41)
		tmp = Float64(l * Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.41)
		tmp = l * -w;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.41], N[(l * (-w)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.409999999999999976

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in w around 0 100.0%

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

   6. Taylor expanded in w around 0 23.4%

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

      1. mul-1-neg 23.4%

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

      2. unsub-neg 23.4%

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

      3. *-commutative 23.4%

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

   8. Simplified 23.4%

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

   9. Taylor expanded in w around inf 23.4%

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

       1. neg-mul-1 23.4%

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

       2. distribute-lft-neg-in 23.4%

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

   11. Simplified 23.4%

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

   if -0.409999999999999976 < w

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 78.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.41:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 99.0%

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

6. Taylor expanded in w around 0 62.7%

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

   1. mul-1-neg 62.7%

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

   2. unsub-neg 62.7%

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

   3. *-commutative 62.7%

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

8. Simplified 62.7%

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

9. Final simplification 62.7%

   \[\leadsto \ell - \ell \cdot w \]
10. Add Preprocessing

Alternative 13: 57.2% 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 99.8%

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

   1. exp-neg 99.8%

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

   2. remove-double-neg 99.8%

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

   3. associate-*l/ 99.8%

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

   4. *-lft-identity 99.8%

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

   5. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in w around 0 57.4%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))