exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 15.2s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Add Preprocessing

Alternative 3: 97.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 82.5%

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

   3. sqr-neg 82.5%

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

   4. sqrt-unprod 42.0%

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

   5. add-sqr-sqrt 82.4%

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

   6. add-sqr-sqrt 82.4%

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

   7. sqrt-unprod 82.4%

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

   8. add-sqr-sqrt 42.0%

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

   9. sqrt-unprod 69.1%

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

   10. sqr-neg 69.1%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 54.7%

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

   13. pow1 54.7%

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

   14. exp-neg 54.7%

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

   15. inv-pow 54.7%

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

   16. pow-prod-up 98.8%

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

   17. metadata-eval 98.8%

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

   18. metadata-eval 98.8%

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

   19. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 97.7% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 82.5%

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

   3. sqr-neg 82.5%

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

   4. sqrt-unprod 42.0%

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

   5. add-sqr-sqrt 82.4%

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

   6. add-sqr-sqrt 82.4%

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

   7. sqrt-unprod 82.4%

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

   8. add-sqr-sqrt 42.0%

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

   9. sqrt-unprod 69.1%

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

   10. sqr-neg 69.1%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 54.7%

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

   13. pow1 54.7%

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

   14. exp-neg 54.7%

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

   15. inv-pow 54.7%

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

   16. pow-prod-up 98.8%

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

   17. metadata-eval 98.8%

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

   18. metadata-eval 98.8%

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

   19. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in w around inf 98.8%

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

   1. exp-neg 98.8%

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

   2. associate-*r/ 98.8%

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

   3. *-rgt-identity 98.8%

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

7. Simplified 98.8%

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

Alternative 5: 77.2% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 - w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666 - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (* l (- 1.0 (* w (+ 1.0 (* w (- (* w 0.16666666666666666) 0.5)))))))

C

double code(double w, double l) {
	return l * (1.0 - (w * (1.0 + (w * ((w * 0.16666666666666666) - 0.5)))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 - (w * (1.0d0 + (w * ((w * 0.16666666666666666d0) - 0.5d0)))))
end function

Java

public static double code(double w, double l) {
	return l * (1.0 - (w * (1.0 + (w * ((w * 0.16666666666666666) - 0.5)))));
}

Python

def code(w, l):
	return l * (1.0 - (w * (1.0 + (w * ((w * 0.16666666666666666) - 0.5)))))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 - Float64(w * Float64(1.0 + Float64(w * Float64(Float64(w * 0.16666666666666666) - 0.5))))))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 - (w * (1.0 + (w * ((w * 0.16666666666666666) - 0.5)))));
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 - N[(w * N[(1.0 + N[(w * N[(N[(w * 0.16666666666666666), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. 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 40.5%

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

   2. sqrt-unprod 82.5%

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

   3. sqr-neg 82.5%

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

   4. sqrt-unprod 42.0%

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

   5. add-sqr-sqrt 82.4%

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

   6. add-sqr-sqrt 82.4%

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

   7. sqrt-unprod 82.4%

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

   8. add-sqr-sqrt 42.0%

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

   9. sqrt-unprod 69.1%

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

   10. sqr-neg 69.1%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 54.7%

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

   13. pow1 54.7%

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

   14. exp-neg 54.7%

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

   15. inv-pow 54.7%

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

   16. pow-prod-up 98.8%

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

   17. metadata-eval 98.8%

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

   18. metadata-eval 98.8%

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

   19. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

   1. frac-2neg 98.8%

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

   2. div-inv 98.8%

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

   3. *-rgt-identity 98.8%

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

8. Applied egg-rr 98.8%

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

9. Taylor expanded in w around 0 79.2%

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

10. Final simplification 79.2%

    \[\leadsto \ell \cdot \left(1 - w \cdot \left(1 + w \cdot \left(w \cdot 0.16666666666666666 - 0.5\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 73.9% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell + \ell \cdot \left(w \cdot \left(w \cdot 0.5 + -1\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* l (* w (+ (* w 0.5) -1.0)))))

C

double code(double w, double l) {
	return l + (l * (w * ((w * 0.5) + -1.0)));
}

Fortran

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

Java

public static double code(double w, double l) {
	return l + (l * (w * ((w * 0.5) + -1.0)));
}

Python

def code(w, l):
	return l + (l * (w * ((w * 0.5) + -1.0)))

Julia

function code(w, l)
	return Float64(l + Float64(l * Float64(w * Float64(Float64(w * 0.5) + -1.0))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (l * (w * ((w * 0.5) + -1.0)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 82.5%

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

   3. sqr-neg 82.5%

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

   4. sqrt-unprod 42.0%

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

   5. add-sqr-sqrt 82.4%

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

   6. add-sqr-sqrt 82.4%

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

   7. sqrt-unprod 82.4%

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

   8. add-sqr-sqrt 42.0%

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

   9. sqrt-unprod 69.1%

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

   10. sqr-neg 69.1%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 54.7%

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

   13. pow1 54.7%

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

   14. exp-neg 54.7%

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

   15. inv-pow 54.7%

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

   16. pow-prod-up 98.8%

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

   17. metadata-eval 98.8%

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

   18. metadata-eval 98.8%

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

   19. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in w around 0 70.0%

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

6. Taylor expanded in l around 0 74.7%

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

7. Final simplification 74.7%

   \[\leadsto \ell + \ell \cdot \left(w \cdot \left(w \cdot 0.5 + -1\right)\right) \]
8. Add Preprocessing

Alternative 7: 70.3% accurate, 33.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 82.5%

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

   3. sqr-neg 82.5%

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

   4. sqrt-unprod 42.0%

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

   5. add-sqr-sqrt 82.4%

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

   6. add-sqr-sqrt 82.4%

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

   7. sqrt-unprod 82.4%

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

   8. add-sqr-sqrt 42.0%

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

   9. sqrt-unprod 69.1%

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

   10. sqr-neg 69.1%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 54.7%

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

   13. pow1 54.7%

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

   14. exp-neg 54.7%

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

   15. inv-pow 54.7%

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

   16. pow-prod-up 98.8%

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

   17. metadata-eval 98.8%

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

   18. metadata-eval 98.8%

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

   19. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in w around 0 70.0%

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

6. Taylor expanded in w around inf 70.0%

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

   1. associate-*r* 70.0%

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

   2. *-commutative 70.0%

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

   3. *-commutative 70.0%

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

   4. *-commutative 70.0%

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

8. Simplified 70.0%

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

9. Final simplification 70.0%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right)\right) \]
10. Add Preprocessing

Alternative 8: 63.3% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 40.5%

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

   2. sqrt-unprod 82.5%

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

   3. sqr-neg 82.5%

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

   4. sqrt-unprod 42.0%

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

   5. add-sqr-sqrt 82.4%

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

   6. add-sqr-sqrt 82.4%

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

   7. sqrt-unprod 82.4%

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

   8. add-sqr-sqrt 42.0%

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

   9. sqrt-unprod 69.1%

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

   10. sqr-neg 69.1%

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

   11. sqrt-unprod 27.1%

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

   12. add-sqr-sqrt 54.7%

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

   13. pow1 54.7%

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

   14. exp-neg 54.7%

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

   15. inv-pow 54.7%

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

   16. pow-prod-up 98.8%

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

   17. metadata-eval 98.8%

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

   18. metadata-eval 98.8%

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

   19. metadata-eval 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in w around 0 63.2%

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

   1. mul-1-neg 63.2%

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

   2. unsub-neg 63.2%

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

7. Simplified 63.2%

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

8. Final simplification 63.2%

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

Alternative 9: 56.6% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in w around 0 56.4%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))