exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 24.1s

Alternatives: 17

Speedup: 1.3×

• could not determine a ground truth

  1. w = 6.213949596521553e-187
  2. l = -9.149067122045835e+283

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

   7. metadata-eval N/A

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

   8. neg-mul-1 N/A

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

   9. exp-neg N/A

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

   10. lift-exp.f64 N/A

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

   11. un-div-inv N/A

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

   12. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ t_1 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow l (exp w))) (t_1 (exp (- w))))
   (if (<= (* t_0 t_1) 5e+306) (/ t_0 (fma w (fma w 0.5 1.0) 1.0)) t_1)))

C

double code(double w, double l) {
	double t_0 = pow(l, exp(w));
	double t_1 = exp(-w);
	double tmp;
	if ((t_0 * t_1) <= 5e+306) {
		tmp = t_0 / fma(w, fma(w, 0.5, 1.0), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = l ^ exp(w)
	t_1 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * t_1) <= 5e+306)
		tmp = Float64(t_0 / fma(w, fma(w, 0.5, 1.0), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * t$95$1), $MachinePrecision], 5e+306], N[(t$95$0 / N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.99999999999999993e306

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. exp-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. un-div-inv N/A

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

      12. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\mathsf{fma}\left(w, 1 + \frac{1}{2} \cdot w, 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(w, \color{blue}{\frac{1}{2} \cdot w + 1}, 1\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(w, \color{blue}{w \cdot \frac{1}{2}} + 1, 1\right)} \]

      5. lower-fma.f64 98.8

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

   9. Applied rewrites 98.8%

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

   if 4.99999999999999993e306 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot \color{blue}{\left({\ell}^{\left(\frac{e^{w}}{2}\right)} \cdot {\ell}^{\left(\frac{e^{w}}{2}\right)}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot \color{blue}{{\ell}^{\left(\frac{e^{w}}{2} + \frac{e^{w}}{2}\right)}} \]

      4. flip-+ N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot {\ell}^{\color{blue}{\left(\frac{\frac{e^{w}}{2} \cdot \frac{e^{w}}{2} - \frac{e^{w}}{2} \cdot \frac{e^{w}}{2}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)}} \]

      5. +-inverses N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot {\ell}^{\left(\frac{\color{blue}{0}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot {\ell}^{\left(\frac{\color{blue}{0 - 0}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot {\ell}^{\left(\frac{\color{blue}{0 \cdot 0} - 0}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(w\right)} \cdot {\ell}^{\left(\frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{\frac{e^{w}}{2} - \frac{e^{w}}{2}}\right)} \]

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \color{blue}{e^{-w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))