exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 11.5s

Alternatives: 11

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{1}{-e^{w}}}\\ \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{e^{-w}}\right)}^{2} \cdot \sqrt[3]{t_0 \cdot \left(t_0 \cdot t_0\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 (- (exp w))))))
   (*
    (- (pow l (exp w)))
    (* (pow (cbrt (exp (- w))) 2.0) (cbrt (* t_0 (* t_0 t_0)))))))

C

double code(double w, double l) {
	double t_0 = cbrt((1.0 / -exp(w)));
	return -pow(l, exp(w)) * (pow(cbrt(exp(-w)), 2.0) * cbrt((t_0 * (t_0 * t_0))));
}

Java

public static double code(double w, double l) {
	double t_0 = Math.cbrt((1.0 / -Math.exp(w)));
	return -Math.pow(l, Math.exp(w)) * (Math.pow(Math.cbrt(Math.exp(-w)), 2.0) * Math.cbrt((t_0 * (t_0 * t_0))));
}

Julia

function code(w, l)
	t_0 = cbrt(Float64(1.0 / Float64(-exp(w))))
	return Float64(Float64(-(l ^ exp(w))) * Float64((cbrt(exp(Float64(-w))) ^ 2.0) * cbrt(Float64(t_0 * Float64(t_0 * t_0)))))
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[N[(1.0 / (-N[Exp[w], $MachinePrecision])), $MachinePrecision], 1/3], $MachinePrecision]}, N[((-N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]) * N[(N[Power[N[Power[N[Exp[(-w)], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. add-cube-cbrt 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]
8. Step-by-step derivation

   1. pow2 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{-e^{w}}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   2. add-sqr-sqrt 29.3%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{\sqrt{-e^{w}} \cdot \sqrt{-e^{w}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   3. sqrt-unprod 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{\sqrt{\left(-e^{w}\right) \cdot \left(-e^{w}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   4. sqr-neg 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\sqrt{\color{blue}{e^{w} \cdot e^{w}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   5. sqrt-unprod 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{\sqrt{e^{w}} \cdot \sqrt{e^{w}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   6. add-sqr-sqrt 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{e^{w}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   7. rec-exp 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\color{blue}{e^{-w}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{e^{-w}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]
10. Step-by-step derivation

    1. add-cube-cbrt 99.7%

       \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]

11. Applied egg-rr 99.7%

    \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{e^{-w}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}}}\right) \]

12. Final simplification 99.7%

    \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{e^{-w}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{-e^{w}}} \cdot \left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)}\right) \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{e^{-w}}\right)}^{2} \cdot \sqrt[3]{\frac{-1}{e^{w}}}\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (*
  (- (pow l (exp w)))
  (* (pow (cbrt (exp (- w))) 2.0) (cbrt (/ -1.0 (exp w))))))

C

double code(double w, double l) {
	return -pow(l, exp(w)) * (pow(cbrt(exp(-w)), 2.0) * cbrt((-1.0 / exp(w))));
}

Java

public static double code(double w, double l) {
	return -Math.pow(l, Math.exp(w)) * (Math.pow(Math.cbrt(Math.exp(-w)), 2.0) * Math.cbrt((-1.0 / Math.exp(w))));
}

Julia

function code(w, l)
	return Float64(Float64(-(l ^ exp(w))) * Float64((cbrt(exp(Float64(-w))) ^ 2.0) * cbrt(Float64(-1.0 / exp(w)))))
end

Wolfram

code[w_, l_] := N[((-N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]) * N[(N[Power[N[Power[N[Exp[(-w)], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[N[(-1.0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. add-cube-cbrt 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]
8. Step-by-step derivation

   1. pow2 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{1}{-e^{w}}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   2. add-sqr-sqrt 29.3%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{\sqrt{-e^{w}} \cdot \sqrt{-e^{w}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   3. sqrt-unprod 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{\sqrt{\left(-e^{w}\right) \cdot \left(-e^{w}\right)}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   4. sqr-neg 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\sqrt{\color{blue}{e^{w} \cdot e^{w}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   5. sqrt-unprod 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{\sqrt{e^{w}} \cdot \sqrt{e^{w}}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   6. add-sqr-sqrt 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{1}{\color{blue}{e^{w}}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   7. rec-exp 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{\color{blue}{e^{-w}}}\right)}^{2} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{e^{-w}}\right)}^{2}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

10. Taylor expanded in w around inf 99.7%

    \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{e^{-w}}\right)}^{2} \cdot \sqrt[3]{\color{blue}{\frac{-1}{e^{w}}}}\right) \]

11. Final simplification 99.7%

    \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\sqrt[3]{e^{-w}}\right)}^{2} \cdot \sqrt[3]{\frac{-1}{e^{w}}}\right) \]

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot {\left(e^{-w}\right)}^{0.6666666666666666}\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (*
  (- (pow l (exp w)))
  (* (cbrt (/ 1.0 (- (exp w)))) (pow (exp (- w)) 0.6666666666666666))))

C

double code(double w, double l) {
	return -pow(l, exp(w)) * (cbrt((1.0 / -exp(w))) * pow(exp(-w), 0.6666666666666666));
}

Java

public static double code(double w, double l) {
	return -Math.pow(l, Math.exp(w)) * (Math.cbrt((1.0 / -Math.exp(w))) * Math.pow(Math.exp(-w), 0.6666666666666666));
}

Julia

function code(w, l)
	return Float64(Float64(-(l ^ exp(w))) * Float64(cbrt(Float64(1.0 / Float64(-exp(w)))) * (exp(Float64(-w)) ^ 0.6666666666666666)))
end

Wolfram

code[w_, l_] := N[((-N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]) * N[(N[Power[N[(1.0 / (-N[Exp[w], $MachinePrecision])), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Exp[(-w)], $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. add-cube-cbrt 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]

7. Applied egg-rr 99.7%

   \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right)} \]
8. Step-by-step derivation

   1. pow1/3 15.2%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\left(\color{blue}{{\left(\frac{1}{-e^{w}}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   2. pow1/3 44.5%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\left({\left(\frac{1}{-e^{w}}\right)}^{0.3333333333333333} \cdot \color{blue}{{\left(\frac{1}{-e^{w}}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   3. pow-prod-up 44.5%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\color{blue}{{\left(\frac{1}{-e^{w}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   4. add-sqr-sqrt 29.3%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\frac{1}{\color{blue}{\sqrt{-e^{w}} \cdot \sqrt{-e^{w}}}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   5. sqrt-unprod 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\frac{1}{\color{blue}{\sqrt{\left(-e^{w}\right) \cdot \left(-e^{w}\right)}}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   6. sqr-neg 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt{\color{blue}{e^{w} \cdot e^{w}}}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   7. sqrt-unprod 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\frac{1}{\color{blue}{\sqrt{e^{w}} \cdot \sqrt{e^{w}}}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   8. add-sqr-sqrt 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(\frac{1}{\color{blue}{e^{w}}}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   9. rec-exp 99.7%

      \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\color{blue}{\left(e^{-w}\right)}}^{\left(0.3333333333333333 + 0.3333333333333333\right)} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

   10. metadata-eval 99.7%

       \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left({\left(e^{-w}\right)}^{\color{blue}{0.6666666666666666}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\color{blue}{{\left(e^{-w}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\frac{1}{-e^{w}}}\right) \]

10. Final simplification 99.7%

    \[\leadsto \left(-{\ell}^{\left(e^{w}\right)}\right) \cdot \left(\sqrt[3]{\frac{1}{-e^{w}}} \cdot {\left(e^{-w}\right)}^{0.6666666666666666}\right) \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

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(Float64(-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. Final simplification 99.7%

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

Alternative 5: 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 6: 97.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in w around 0 98.1%

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

3. Final simplification 98.1%

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

Alternative 7: 97.6% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 98.1%

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

5. Final simplification 98.1%

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

Alternative 8: 64.2% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.56:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.56) (* l (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.56) {
		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 <= (-0.56d0)) 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 <= -0.56) {
		tmp = l * -w;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.56:
		tmp = l * -w
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.56)
		tmp = Float64(l * Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.56)
		tmp = l * -w;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.56], N[(l * (-w)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.56000000000000005

   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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around 0 5.5%

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

   7. Taylor expanded in w around 0 29.1%

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

   8. Taylor expanded in w around inf 29.1%

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

      1. associate-*r* 29.1%

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

      2. mul-1-neg 29.1%

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

   10. Simplified 29.1%

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

   if -0.56000000000000005 < w

   1. Initial program 99.6%

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

      1. exp-neg 99.6%

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

      2. associate-*l/ 99.6%

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

      3. *-lft-identity 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in w around 0 76.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.56:\\ \;\;\;\;\ell \cdot \left(-w\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 9: 63.9% accurate, 50.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in w around 0 71.4%

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

7. Taylor expanded in w around 0 62.6%

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

8. Final simplification 62.6%

   \[\leadsto \left(-\ell\right) \cdot \left(w + -1\right) \]

Alternative 10: 63.9% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 98.1%

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

5. Taylor expanded in w around 0 62.6%

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

   1. +-commutative 62.6%

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

   2. mul-1-neg 62.6%

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

   3. unsub-neg 62.6%

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

   4. *-commutative 62.6%

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

7. Simplified 62.6%

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

8. Final simplification 62.6%

   \[\leadsto \ell - \ell \cdot w \]

Alternative 11: 57.4% 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 99.7%

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

   1. exp-neg 99.7%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in w around 0 55.5%

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

7. Final simplification 55.5%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023200 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))