exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 10.5s

Alternatives: 5

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. remove-double-neg 99.5%

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

   3. associate-*l/ 99.5%

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

   4. *-lft-identity 99.5%

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

   5. remove-double-neg 99.5%

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

3. Simplified 99.5%

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

Alternative 3: 78.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -120000000 \lor \neg \left(w \leq 0.105\right):\\ \;\;\;\;0.03333333333333333 \cdot {\ell}^{5}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -120000000.0) (not (<= w 0.105)))
   (* 0.03333333333333333 (pow l 5.0))
   l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -120000000.0) || !(w <= 0.105)) {
		tmp = 0.03333333333333333 * pow(l, 5.0);
	} 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 <= (-120000000.0d0)) .or. (.not. (w <= 0.105d0))) then
        tmp = 0.03333333333333333d0 * (l ** 5.0d0)
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -120000000.0) || !(w <= 0.105)) {
		tmp = 0.03333333333333333 * Math.pow(l, 5.0);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -120000000.0) or not (w <= 0.105):
		tmp = 0.03333333333333333 * math.pow(l, 5.0)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -120000000.0) || !(w <= 0.105))
		tmp = Float64(0.03333333333333333 * (l ^ 5.0));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -120000000.0) || ~((w <= 0.105)))
		tmp = 0.03333333333333333 * (l ^ 5.0);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -120000000.0], N[Not[LessEqual[w, 0.105]], $MachinePrecision]], N[(0.03333333333333333 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -1.2e8 or 0.104999999999999996 < w

   1. Initial program 99.1%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.1%

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

      3. add-sqr-sqrt 62.4%

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

      4. sqrt-unprod 64.1%

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

      5. sqr-neg 64.1%

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

      6. sqrt-unprod 1.7%

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

      7. add-sqr-sqrt 1.7%

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

      8. pow1 1.7%

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

      9. exp-neg 1.7%

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

      10. inv-pow 1.7%

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

      11. pow-prod-up 39.6%

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

      12. metadata-eval 39.6%

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

      13. metadata-eval 39.6%

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

      14. metadata-eval 39.6%

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

      15. *-un-lft-identity 39.6%

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

      16. add-sqr-sqrt 37.6%

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

      17. sqrt-unprod 66.3%

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

      18. sqr-neg 66.3%

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

      19. sqrt-unprod 28.7%

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

      20. add-sqr-sqrt 29.9%

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

   4. Applied egg-rr 31.9%

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

   5. Taylor expanded in l around 0 20.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\ell \cdot \left(1 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-commutative 20.3%

         \[\leadsto \mathsf{log1p}\left(\ell \cdot \left(1 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right) \]

   7. Simplified 20.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}\right) \]

   8. Taylor expanded in l around 0 23.1%

      \[\leadsto \color{blue}{\ell \cdot \left(1 + {\ell}^{3} \cdot \left(0.03333333333333333 \cdot \ell - 0.041666666666666664\right)\right)} \]

   9. Taylor expanded in l around inf 50.5%

      \[\leadsto \color{blue}{0.03333333333333333 \cdot {\ell}^{5}} \]

   if -1.2e8 < w < 0.104999999999999996

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0 97.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -120000000 \lor \neg \left(w \leq 0.105\right):\\ \;\;\;\;0.03333333333333333 \cdot {\ell}^{5}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 48.0%

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

   2. sqrt-unprod 83.0%

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

   3. sqr-neg 83.0%

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

   4. sqrt-unprod 35.0%

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

   5. add-sqr-sqrt 82.0%

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

   6. add-sqr-sqrt 82.0%

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

   7. sqrt-unprod 82.0%

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

   8. add-sqr-sqrt 35.0%

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

   9. sqrt-unprod 66.0%

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

   10. sqr-neg 66.0%

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

   11. sqrt-unprod 31.0%

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

   12. add-sqr-sqrt 53.1%

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

   13. pow1 53.1%

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

   14. exp-neg 53.1%

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

   15. inv-pow 53.1%

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

   16. pow-prod-up 98.4%

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

   17. metadata-eval 98.4%

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

   18. metadata-eval 98.4%

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

   19. metadata-eval 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 56.9% 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 55.1%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))