exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 17.4s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. w = 2.3987730335762524e+71
  2. l = 3.8655961973589113e+207

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 98.7%

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

Alternative 2: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot {\ell}^{\left(e^{w}\right)} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* t_0 (pow l (exp w))) 2e+297)
     (* (- 1.0 w) (pow l (fma (fma 0.5 w 1.0) w 1.0)))
     t_0)))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((t_0 * pow(l, exp(w))) <= 2e+297) {
		tmp = (1.0 - w) * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * (l ^ exp(w))) <= 2e+297)
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e+297], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 2e297

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.3

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in w around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 98.6

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

   11. Applied rewrites 98.6%

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

   if 2e297 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 97.1%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 97.2

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

   4. Applied rewrites 97.2%

      \[\leadsto e^{-w} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1:\\ \;\;\;\;1 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.0)
   (* 1.0 (pow l (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (* (fma (fma 0.5 w -1.0) w 1.0) (pow l (fma (fma 0.5 w 1.0) w 1.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if l < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in w around 0

      \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(w \cdot \left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto 1 \cdot {\ell}^{\left(\color{blue}{\left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right) \cdot w} + 1\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(\mathsf{fma}\left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right), w, 1\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto 1 \cdot {\ell}^{\left(\mathsf{fma}\left(\color{blue}{w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right) + 1}, w, 1\right)\right)} \]

      5. *-commutative N/A

         \[\leadsto 1 \cdot {\ell}^{\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot w\right) \cdot w} + 1, w, 1\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto 1 \cdot {\ell}^{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot w, w, 1\right)}, w, 1\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 1 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot w + \frac{1}{2}}, w, 1\right), w, 1\right)\right)} \]

      8. lower-fma.f64 98.1

         \[\leadsto 1 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, w, 0.5\right)}, w, 1\right), w, 1\right)\right)} \]

   8. Applied rewrites 98.1%

      \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)\right)}} \]

   if 1 < l

   1. Initial program 97.7%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right) \cdot {\ell}^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3) (exp (- w)) (* 1.0 (pow l (fma (fma 0.5 w 1.0) w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = exp(-w);
	} else {
		tmp = 1.0 * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.3)
		tmp = exp(Float64(-w));
	else
		tmp = Float64(1.0 * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[Exp[(-w)], $MachinePrecision], N[(1.0 * N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.30000000000000004

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   if -1.30000000000000004 < w

   1. Initial program 98.3%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.4

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

   8. Applied rewrites 97.4%

      \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.4 \cdot 10^{+16}:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(1 + w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.4e+16) (exp (- w)) (* 1.0 (pow l (+ 1.0 w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4.4e+16) {
		tmp = exp(-w);
	} else {
		tmp = 1.0 * pow(l, (1.0 + w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-4.4d+16)) then
        tmp = exp(-w)
    else
        tmp = 1.0d0 * (l ** (1.0d0 + w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -4.4e+16) {
		tmp = Math.exp(-w);
	} else {
		tmp = 1.0 * Math.pow(l, (1.0 + w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -4.4e+16:
		tmp = math.exp(-w)
	else:
		tmp = 1.0 * math.pow(l, (1.0 + w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4.4e+16)
		tmp = exp(Float64(-w));
	else
		tmp = Float64(1.0 * (l ^ Float64(1.0 + w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -4.4e+16)
		tmp = exp(-w);
	else
		tmp = 1.0 * (l ^ (1.0 + w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4.4e+16], N[Exp[(-w)], $MachinePrecision], N[(1.0 * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4.4e16

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   if -4.4e16 < w

   1. Initial program 98.3%

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

   3. Taylor expanded in w around 0

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 97.4

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

   8. Applied rewrites 97.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.4 \cdot 10^{+16}:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(1 + w\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

       \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

   13. metadata-eval 46.3

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

4. Applied rewrites 46.3%

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

5. Final simplification 46.3%

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

Alternative 7: 26.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(w \cdot w - 1\right) \cdot \left(1 - w\right) - \left(w - 1\right) \cdot \left(\left(1 - w \cdot w\right) \cdot \left(w \cdot w\right)\right)}{\left(w - 1\right) \cdot \left(1 - w\right)}}{\left(1 + w\right) \cdot \left(1 + w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -5.5e+102)
   (fma (fma (fma -0.16666666666666666 w 0.5) w -1.0) w 1.0)
   (/
    (/
     (-
      (* (- (* w w) 1.0) (- 1.0 w))
      (* (- w 1.0) (* (- 1.0 (* w w)) (* w w))))
     (* (- w 1.0) (- 1.0 w)))
    (* (+ 1.0 w) (+ 1.0 w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -5.5e+102) {
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	} else {
		tmp = (((((w * w) - 1.0) * (1.0 - w)) - ((w - 1.0) * ((1.0 - (w * w)) * (w * w)))) / ((w - 1.0) * (1.0 - w))) / ((1.0 + w) * (1.0 + w));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -5.5e+102)
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(w * w) - 1.0) * Float64(1.0 - w)) - Float64(Float64(w - 1.0) * Float64(Float64(1.0 - Float64(w * w)) * Float64(w * w)))) / Float64(Float64(w - 1.0) * Float64(1.0 - w))) / Float64(Float64(1.0 + w) * Float64(1.0 + w)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -5.5e+102], N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w + -1.0), $MachinePrecision] * w + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(w * w), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision] - N[(N[(w - 1.0), $MachinePrecision] * N[(N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(w - 1.0), $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + w), $MachinePrecision] * N[(1.0 + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -5.49999999999999981e102

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)} \cdot 1 \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) + 1\right)} \cdot 1 \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) \cdot w} + 1\right) \cdot 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1, w, 1\right)} \cdot 1 \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)}, w, 1\right) \cdot 1 \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) \cdot w} + \left(\mathsf{neg}\left(1\right)\right), w, 1\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) \cdot w + \color{blue}{-1}, w, 1\right) \cdot 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot w, w, -1\right)}, w, 1\right) \cdot 1 \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot w + \frac{1}{2}}, w, -1\right), w, 1\right) \cdot 1 \]

      9. lower-fma.f64 98.5

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right)}, w, -1\right), w, 1\right) \cdot 1 \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)} \cdot 1 \]

   if -5.49999999999999981e102 < w

   1. Initial program 98.4%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 32.6

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

   4. Applied rewrites 32.6%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 4.4

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 4.2%

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

   11. Applied rewrites 7.9%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(w \cdot w - 1\right) \cdot \left(1 - w\right) - \left(w - 1\right) \cdot \left(\left(1 - w \cdot w\right) \cdot \left(w \cdot w\right)\right)}{\left(w - 1\right) \cdot \left(1 - w\right)}}{\left(1 + w\right) \cdot \left(1 + w\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - w \cdot w\\ \mathbf{if}\;w \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0 - t\_0 \cdot \left(w \cdot w\right)}{1 - w}}{\left(1 + w\right) \cdot \left(1 + w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- 1.0 (* w w))))
   (if (<= w -1.35e+154)
     (fma (fma 0.5 w -1.0) w 1.0)
     (/ (/ (- t_0 (* t_0 (* w w))) (- 1.0 w)) (* (+ 1.0 w) (+ 1.0 w))))))

C

double code(double w, double l) {
	double t_0 = 1.0 - (w * w);
	double tmp;
	if (w <= -1.35e+154) {
		tmp = fma(fma(0.5, w, -1.0), w, 1.0);
	} else {
		tmp = ((t_0 - (t_0 * (w * w))) / (1.0 - w)) / ((1.0 + w) * (1.0 + w));
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = Float64(1.0 - Float64(w * w))
	tmp = 0.0
	if (w <= -1.35e+154)
		tmp = fma(fma(0.5, w, -1.0), w, 1.0);
	else
		tmp = Float64(Float64(Float64(t_0 - Float64(t_0 * Float64(w * w))) / Float64(1.0 - w)) / Float64(Float64(1.0 + w) * Float64(1.0 + w)));
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.35e+154], N[(N[(0.5 * w + -1.0), $MachinePrecision] * w + 1.0), $MachinePrecision], N[(N[(N[(t$95$0 - N[(t$95$0 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - w), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + w), $MachinePrecision] * N[(1.0 + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.35000000000000003e154

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, w, -1\right)}, w, 1\right) \cdot 1 \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right)} \cdot 1 \]

   if -1.35000000000000003e154 < w

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. flip-- N/A

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

      12. metadata-eval N/A

          \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

      13. metadata-eval 36.4

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

   4. Applied rewrites 36.4%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 4.4

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 9.5%

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

   11. Applied rewrites 11.3%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - w \cdot w\right) - \left(1 - w \cdot w\right) \cdot \left(w \cdot w\right)}{1 - w}}{\left(1 + w\right) \cdot \left(1 + w\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 22.6% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right) \end{array} \]

FPCore

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

C

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

Julia

function code(w, l)
	return fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0)
end

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

       \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

   13. metadata-eval 46.3

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

4. Applied rewrites 46.3%

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

5. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)} \cdot 1 \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) + 1\right)} \cdot 1 \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) \cdot w} + 1\right) \cdot 1 \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1, w, 1\right)} \cdot 1 \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)}, w, 1\right) \cdot 1 \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) \cdot w} + \left(\mathsf{neg}\left(1\right)\right), w, 1\right) \cdot 1 \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) \cdot w + \color{blue}{-1}, w, 1\right) \cdot 1 \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{6} \cdot w, w, -1\right)}, w, 1\right) \cdot 1 \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot w + \frac{1}{2}}, w, -1\right), w, 1\right) \cdot 1 \]

   9. lower-fma.f64 23.6

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right)}, w, -1\right), w, 1\right) \cdot 1 \]

7. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)} \cdot 1 \]

8. Final simplification 23.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right) \]
9. Add Preprocessing

Alternative 10: 18.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (fma (fma 0.5 w -1.0) w 1.0))

C

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

Julia

function code(w, l)
	return fma(fma(0.5, w, -1.0), w, 1.0)
end

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

       \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

   13. metadata-eval 46.3

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

4. Applied rewrites 46.3%

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

5. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 19.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, w, -1\right)}, w, 1\right) \cdot 1 \]

7. Applied rewrites 19.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right)} \cdot 1 \]

8. Final simplification 19.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, w, -1\right), w, 1\right) \]
9. Add Preprocessing

Alternative 11: 4.9% accurate, 77.3× speedup

Math

\[\begin{array}{l} \\ 1 - w \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

       \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

   13. metadata-eval 46.3

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

4. Applied rewrites 46.3%

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

5. Taylor expanded in w around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 4.8

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

7. Applied rewrites 4.8%

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

8. Final simplification 4.8%

   \[\leadsto 1 - w \]
9. Add Preprocessing

Alternative 12: 4.0% accurate, 103.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return -w

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := (-w)

Derivation

1. Initial program 98.7%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. flip-- N/A

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

   12. metadata-eval N/A

       \[\leadsto e^{-w} \cdot {\ell}^{\color{blue}{0}} \]

   13. metadata-eval 46.3

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

4. Applied rewrites 46.3%

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

5. Taylor expanded in w around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 4.8

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

7. Applied rewrites 4.8%

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

8. Taylor expanded in w around inf

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

10. Applied rewrites 3.9%

    \[\leadsto \left(-w\right) \cdot 1 \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-rgt-identity 3.9

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

12. Applied rewrites 3.9%

    \[\leadsto \color{blue}{-w} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024307 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))