exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 14.9s

Alternatives: 5

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = -8.895527935071599e+59
  2. l = -1.9976865686336553e+154

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

Alternative 2: 98.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.0009:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.0009)
   (*
    (- 1.0 w)
    (pow l (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (* (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 <= 0.0009) {
		tmp = (1.0 - w) * pow(l, fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	} else {
		tmp = 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 <= 0.0009)
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)));
	else
		tmp = Float64(fma(Float64(Float64(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, 0.0009], N[(N[(1.0 - w), $MachinePrecision] * 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[(N[(0.5 * w), $MachinePrecision] - 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 < 8.9999999999999998e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. metadata-eval N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 8.9999999999999998e-4 < l

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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. lower--.f64 N/A

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

      5. lower-*.f64 77.3

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.4

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

   8. Applied rewrites 98.4%

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

Alternative 3: 98.7% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.0)
   (*
    (- 1.0 w)
    (pow l (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (* 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 - w) * pow(l, fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	} 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 (l <= 1.0)
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)));
	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[l, 1.0], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $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 l < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. metadata-eval N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.2

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

   8. Applied rewrites 99.2%

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

   if 1 < l

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 54.8%

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

   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 97.9

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

   8. Applied rewrites 97.9%

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

Alternative 4: 98.6% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.0)
   (* (- 1.0 w) (pow l (+ 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 - w) * pow(l, (1.0 + 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 (l <= 1.0)
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(1.0 + 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[l, 1.0], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $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 l < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. *-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. metadata-eval N/A

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

      6. *-rgt-identity N/A

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

      7. lower--.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

   if 1 < l

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 54.8%

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

   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 97.9

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

   8. Applied rewrites 97.9%

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

Alternative 5: 85.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return 1.0 * math.pow(l, (1.0 + w))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 63.9%

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

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 79.1

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

8. Applied rewrites 79.1%

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

Reproduce

herbie shell --seed 2024343 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))