exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 17.2s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. w = -1.980106633632649e-263
  2. l = -9.044523106380322e-304

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

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

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 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ t_1 := {\ell}^{\left(e^{w}\right)} \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;{\ell}^{1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))) (t_1 (* (pow l (exp w)) t_0)))
   (if (<= t_1 0.0) t_0 (if (<= t_1 INFINITY) (* (pow l 1.0) 1.0) t_0))))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double t_1 = pow(l, exp(w)) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = pow(l, 1.0) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.exp(-w);
	double t_1 = Math.pow(l, Math.exp(w)) * t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow(l, 1.0) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.exp(-w)
	t_1 = math.pow(l, math.exp(w)) * t_0
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= math.inf:
		tmp = math.pow(l, 1.0) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	t_1 = Float64((l ^ exp(w)) * t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64((l ^ 1.0) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = exp(-w);
	t_1 = (l ^ exp(w)) * t_0;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = (l ^ 1.0) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[Power[l, 1.0], $MachinePrecision] * 1.0), $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))) < 0.0 or +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 100.0

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

   6. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < +inf.0

   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 68.1%

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

   6. Taylor expanded in w around 0

      \[\leadsto 1 \cdot {\ell}^{\color{blue}{1}} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 0:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq \infty:\\ \;\;\;\;{\ell}^{1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{-w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ t_1 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot t\_1 \leq \infty:\\ \;\;\;\;\left(1 - w\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow l (exp w))) (t_1 (exp (- w))))
   (if (<= (* t_0 t_1) INFINITY) (* (- 1.0 w) t_0) t_1)))

C

double code(double w, double l) {
	double t_0 = pow(l, exp(w));
	double t_1 = exp(-w);
	double tmp;
	if ((t_0 * t_1) <= ((double) INFINITY)) {
		tmp = (1.0 - w) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.pow(l, Math.exp(w));
	double t_1 = Math.exp(-w);
	double tmp;
	if ((t_0 * t_1) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 - w) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.pow(l, math.exp(w))
	t_1 = math.exp(-w)
	tmp = 0
	if (t_0 * t_1) <= math.inf:
		tmp = (1.0 - w) * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(w, l)
	t_0 = l ^ exp(w)
	t_1 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * t_1) <= Inf)
		tmp = Float64(Float64(1.0 - w) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = l ^ exp(w);
	t_1 = exp(-w);
	tmp = 0.0;
	if ((t_0 * t_1) <= Inf)
		tmp = (1.0 - w) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < +inf.0

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 44.1

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

   6. Applied rewrites 44.1%

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

4. Final simplification 74.2%

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

Alternative 4: 84.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((pow(l, exp(w)) * t_0) <= ((double) INFINITY)) {
		tmp = pow(l, fma(fma(0.5, w, 1.0), w, 1.0)) * fma(-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((l ^ exp(w)) * t_0) <= Inf)
		tmp = Float64((l ^ fma(fma(0.5, w, 1.0), w, 1.0)) * fma(-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[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * w + 1.0), $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))) < +inf.0

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   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 80.2

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 88.7%

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

   if +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 44.1

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

   6. Applied rewrites 44.1%

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

4. Final simplification 88.7%

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

Alternative 5: 84.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((pow(l, exp(w)) * t_0) <= ((double) INFINITY)) {
		tmp = pow(l, (1.0 + w)) * fma(-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((l ^ exp(w)) * t_0) <= Inf)
		tmp = Float64((l ^ Float64(1.0 + w)) * fma(-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[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision] * N[(-1.0 * w + 1.0), $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))) < +inf.0

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 74.7

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 83.4%

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

   if +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 44.1

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

   6. Applied rewrites 44.1%

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

4. Final simplification 83.4%

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

Alternative 6: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot t\_0 \leq \infty:\\ \;\;\;\;1 \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* (pow l (exp w)) t_0) INFINITY) (* 1.0 (pow l (+ 1.0 w))) t_0)))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((pow(l, exp(w)) * t_0) <= ((double) INFINITY)) {
		tmp = 1.0 * pow(l, (1.0 + w));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.exp(-w);
	double tmp;
	if ((Math.pow(l, Math.exp(w)) * t_0) <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 * Math.pow(l, (1.0 + w));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.exp(-w)
	tmp = 0
	if (math.pow(l, math.exp(w)) * t_0) <= math.inf:
		tmp = 1.0 * math.pow(l, (1.0 + w))
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(w, l)
	t_0 = exp(-w);
	tmp = 0.0;
	if (((l ^ exp(w)) * t_0) <= Inf)
		tmp = 1.0 * (l ^ (1.0 + w));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], Infinity], N[(1.0 * N[Power[l, N[(1.0 + w), $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))) < +inf.0

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 72.8%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if +inf.0 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 44.1

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

   4. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 44.1

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

   6. Applied rewrites 44.1%

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

4. Final simplification 82.8%

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

Alternative 7: 25.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\mathsf{fma}\left(w \cdot w, -2, 1\right)}{\left(1 - w \cdot w\right) \cdot \left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= (* (pow l (exp w)) (exp (- w))) 5e-93)
   (/ (fma (* w w) -2.0 1.0) (* (- 1.0 (* w w)) (+ 1.0 w)))
   (fma (fma (fma -0.16666666666666666 w 0.5) w -1.0) w 1.0)))

C

double code(double w, double l) {
	double tmp;
	if ((pow(l, exp(w)) * exp(-w)) <= 5e-93) {
		tmp = fma((w * w), -2.0, 1.0) / ((1.0 - (w * w)) * (1.0 + w));
	} else {
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (Float64((l ^ exp(w)) * exp(Float64(-w))) <= 5e-93)
		tmp = Float64(fma(Float64(w * w), -2.0, 1.0) / Float64(Float64(1.0 - Float64(w * w)) * Float64(1.0 + w)));
	else
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[Exp[(-w)], $MachinePrecision]), $MachinePrecision], 5e-93], N[(N[(N[(w * w), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(1.0 + w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w + -1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.99999999999999994e-93

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 48.6

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

   4. Applied rewrites 48.6%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - w} \]

      3. lower--.f64 3.3

         \[\leadsto \color{blue}{1 - w} \]

   7. Applied rewrites 3.3%

      \[\leadsto \color{blue}{1 - w} \]
   8. Step-by-step derivation

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 10.0%

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

   if 4.99999999999999994e-93 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 28.9

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

   7. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\mathsf{fma}\left(w \cdot w, -2, 1\right)}{\left(1 - w \cdot w\right) \cdot \left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.7% 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 44.1

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

4. Applied rewrites 44.1%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-exp.f64 44.1

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

6. Applied rewrites 44.1%

   \[\leadsto \color{blue}{e^{-w}} \]
7. Add Preprocessing

Alternative 9: 27.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - w \cdot w\right) \cdot \left(1 + w\right)\\ \mathbf{if}\;w \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right), w, -1\right), w, 1\right)\\ \mathbf{elif}\;w \leq -1 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(w, w, -2\right), w \cdot w, 1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(w \cdot w, -2, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (* w w)) (+ 1.0 w))))
   (if (<= w -5.6e+102)
     (fma (fma (fma -0.16666666666666666 w 0.5) w -1.0) w 1.0)
     (if (<= w -1e-42)
       (/ (fma (fma w w -2.0) (* w w) 1.0) t_0)
       (/ (fma (* w w) -2.0 1.0) t_0)))))

C

double code(double w, double l) {
	double t_0 = (1.0 - (w * w)) * (1.0 + w);
	double tmp;
	if (w <= -5.6e+102) {
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	} else if (w <= -1e-42) {
		tmp = fma(fma(w, w, -2.0), (w * w), 1.0) / t_0;
	} else {
		tmp = fma((w * w), -2.0, 1.0) / t_0;
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = Float64(Float64(1.0 - Float64(w * w)) * Float64(1.0 + w))
	tmp = 0.0
	if (w <= -5.6e+102)
		tmp = fma(fma(fma(-0.16666666666666666, w, 0.5), w, -1.0), w, 1.0);
	elseif (w <= -1e-42)
		tmp = Float64(fma(fma(w, w, -2.0), Float64(w * w), 1.0) / t_0);
	else
		tmp = Float64(fma(Float64(w * w), -2.0, 1.0) / t_0);
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(1.0 - N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(1.0 + w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -5.6e+102], N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w + -1.0), $MachinePrecision] * w + 1.0), $MachinePrecision], If[LessEqual[w, -1e-42], N[(N[(N[(w * w + -2.0), $MachinePrecision] * N[(w * w), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(w * w), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if w < -5.60000000000000037e102

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -5.60000000000000037e102 < w < -1.00000000000000004e-42

   1. Initial program 98.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 60.8

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - w} \]

      3. lower--.f64 4.4

         \[\leadsto \color{blue}{1 - w} \]

   7. Applied rewrites 4.4%

      \[\leadsto \color{blue}{1 - w} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.5%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 18.5%

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

   if -1.00000000000000004e-42 < w

   1. Initial program 99.7%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 26.1

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

   4. Applied rewrites 26.1%

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

   5. Taylor expanded in w around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

         \[\leadsto \color{blue}{1 - w} \]

      3. lower--.f64 4.5

         \[\leadsto \color{blue}{1 - w} \]

   7. Applied rewrites 4.5%

      \[\leadsto \color{blue}{1 - w} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.2%

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

   10. Taylor expanded in w around 0

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

   12. Applied rewrites 7.7%

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

Alternative 10: 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 44.1

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 20.6

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

7. Applied rewrites 20.6%

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

Alternative 11: 18.1% 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 44.1

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. lft-mult-inverse N/A

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

   5. distribute-lft-neg-out N/A

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

   6. distribute-rgt-in N/A

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

   7. sub-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 14.8

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

7. Applied rewrites 14.8%

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

Alternative 12: 18.1% accurate, 25.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(w, l)
	return fma(Float64(0.5 * w), w, 1.0)
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 44.1

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. lft-mult-inverse N/A

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

   5. distribute-lft-neg-out N/A

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

   6. distribute-rgt-in N/A

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

   7. sub-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 14.8

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

7. Applied rewrites 14.8%

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

8. Taylor expanded in w around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot w, w, 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 14.8%

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

Alternative 13: 17.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \left(w \cdot w\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (* w w) 0.5))

C

double code(double w, double l) {
	return (w * w) * 0.5;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (w * w) * 0.5d0
end function

Java

public static double code(double w, double l) {
	return (w * w) * 0.5;
}

Python

def code(w, l):
	return (w * w) * 0.5

Julia

function code(w, l)
	return Float64(Float64(w * w) * 0.5)
end

MATLAB

function tmp = code(w, l)
	tmp = (w * w) * 0.5;
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 44.1

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub-neg N/A

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

   4. lft-mult-inverse N/A

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

   5. distribute-lft-neg-out N/A

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

   6. distribute-rgt-in N/A

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

   7. sub-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. distribute-rgt-in N/A

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

   11. distribute-lft-neg-out N/A

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

   12. lft-mult-inverse N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 14.8

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

7. Applied rewrites 14.8%

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

8. Taylor expanded in w around inf

   \[\leadsto \frac{1}{2} \cdot \color{blue}{{w}^{2}} \]
9. Step-by-step derivation

10. Applied rewrites 14.2%

    \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{0.5} \]
11. Add Preprocessing

Alternative 14: 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 44.1

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in w around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{1 - w} \]

   3. lower--.f64 4.7

      \[\leadsto \color{blue}{1 - w} \]

7. Applied rewrites 4.7%

   \[\leadsto \color{blue}{1 - w} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))