exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 21.5s

Alternatives: 18

Speedup: 1.0×

• could not determine a ground truth

  1. w = -1.5614527825830153e-159
  2. l = -3.283472033018188e+263

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

\[\begin{array}{l} \\ {\ell}^{\left(e^{w}\right)} \cdot e^{0 - w} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (pow l (exp w)) (exp (- 0.0 w))))

C

double code(double w, double l) {
	return pow(l, exp(w)) * exp((0.0 - w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (l ** exp(w)) * exp((0.0d0 - w))
end function

Java

public static double code(double w, double l) {
	return Math.pow(l, Math.exp(w)) * Math.exp((0.0 - w));
}

Python

def code(w, l):
	return math.pow(l, math.exp(w)) * math.exp((0.0 - w))

Julia

function code(w, l)
	return Float64((l ^ exp(w)) * exp(Float64(0.0 - w)))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) * exp((0.0 - w));
end

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

   \[\leadsto {\ell}^{\left(e^{w}\right)} \cdot e^{0 - w} \]
4. Add Preprocessing

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 3.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 3.1e-17) (/ l (exp w)) (exp (- (* (exp w) (log l)) w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 3.1e-17) {
		tmp = l / exp(w);
	} else {
		tmp = exp(((exp(w) * log(l)) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 3.1d-17) then
        tmp = l / exp(w)
    else
        tmp = exp(((exp(w) * log(l)) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 3.1e-17) {
		tmp = l / Math.exp(w);
	} else {
		tmp = Math.exp(((Math.exp(w) * Math.log(l)) - w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 3.1e-17:
		tmp = l / math.exp(w)
	else:
		tmp = math.exp(((math.exp(w) * math.log(l)) - w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 3.1e-17)
		tmp = Float64(l / exp(w));
	else
		tmp = exp(Float64(Float64(exp(w) * log(l)) - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 3.1e-17)
		tmp = l / exp(w);
	else
		tmp = exp(((exp(w) * log(l)) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 3.1e-17], N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Exp[w], $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.0999999999999998e-17

   1. Initial program 99.9%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.3%

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

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(e^{w}\right)}\right) \]

      5. exp-lowering-exp.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right) \]

   7. Applied egg-rr 99.3%

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

   if 3.0999999999999998e-17 < w

   1. Initial program 98.7%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \ell \cdot e^{w}\right), w\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \ell, \left(e^{w}\right)\right), w\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \left(e^{w}\right)\right), w\right)\right) \]

      4. exp-lowering-exp.f64 98.8%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\ell\right), \mathsf{exp.f64}\left(w\right)\right), w\right)\right) \]

   7. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.5:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot \left(1 - w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.5) (exp (- 0.0 w)) (* (pow l (exp w)) (- 1.0 w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4.5) {
		tmp = exp((0.0 - w));
	} else {
		tmp = pow(l, exp(w)) * (1.0 - w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-4.5d0)) then
        tmp = exp((0.0d0 - w))
    else
        tmp = (l ** exp(w)) * (1.0d0 - w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -4.5) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = Math.pow(l, Math.exp(w)) * (1.0 - w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -4.5:
		tmp = math.exp((0.0 - w))
	else:
		tmp = math.pow(l, math.exp(w)) * (1.0 - w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4.5)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64((l ^ exp(w)) * Float64(1.0 - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -4.5)
		tmp = exp((0.0 - w));
	else
		tmp = (l ^ exp(w)) * (1.0 - w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4.5], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4.5

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -4.5 < w

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.5:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{\left(e^{w}\right)} \cdot \left(1 - w\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1500.0)
   (exp (- 0.0 w))
   (*
    (- 1.0 w)
    (pow l (fma w (fma w (fma w 0.16666666666666666 0.5) 1.0) 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = exp((0.0 - w));
	} else {
		tmp = (1.0 - w) * pow(l, fma(w, fma(w, fma(w, 0.16666666666666666, 0.5), 1.0), 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1500.0)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(w, fma(w, fma(w, 0.16666666666666666, 0.5), 1.0), 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1500.0], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w * N[(w * N[(w * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1500

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -1500 < w

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \left(w \cdot \left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right) + \color{blue}{1}\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \color{blue}{\left(1 + w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right)\right)}, 1\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot w\right) + \color{blue}{1}\right), 1\right)\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot w\right)}, 1\right), 1\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(\frac{1}{6} \cdot w + \color{blue}{\frac{1}{2}}\right), 1\right), 1\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(w \cdot \frac{1}{6} + \frac{1}{2}\right), 1\right), 1\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \color{blue}{\frac{1}{6}}, \frac{1}{2}\right), 1\right), 1\right)\right)\right) \]

   8. Simplified 98.8%

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

Alternative 5: 98.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, 1\right), 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3)
   (exp (- 0.0 w))
   (* (- 1.0 w) (pow l (fma w (fma w 0.5 1.0) 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = exp((0.0 - w));
	} else {
		tmp = (1.0 - w) * pow(l, fma(w, fma(w, 0.5, 1.0), 1.0));
	}
	return tmp;
}

Julia

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w * N[(w * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -1.30000000000000004 < w

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \left(w \cdot \left(1 + \frac{1}{2} \cdot w\right) + \color{blue}{1}\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \color{blue}{\left(1 + \frac{1}{2} \cdot w\right)}, 1\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \left(\frac{1}{2} \cdot w + \color{blue}{1}\right), 1\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \left(w \cdot \frac{1}{2} + 1\right), 1\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \color{blue}{\frac{1}{2}}, 1\right), 1\right)\right)\right) \]

   8. Simplified 98.8%

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

Alternative 6: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot \left(\ell \cdot {\ell}^{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1500.0) (exp (- 0.0 w)) (* (- 1.0 w) (* l (pow l w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = exp((0.0 - w));
	} else {
		tmp = (1.0 - w) * (l * pow(l, w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1500.0d0)) then
        tmp = exp((0.0d0 - w))
    else
        tmp = (1.0d0 - w) * (l * (l ** w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = (1.0 - w) * (l * Math.pow(l, w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1500.0:
		tmp = math.exp((0.0 - w))
	else:
		tmp = (1.0 - w) * (l * math.pow(l, w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1500.0)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64(Float64(1.0 - w) * Float64(l * (l ^ w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1500.0)
		tmp = exp((0.0 - w));
	else
		tmp = (1.0 - w) * (l * (l ^ w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1500.0], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[(l * N[Power[l, w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1500

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -1500 < w

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \color{blue}{\left(1 + w\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{w}\right)\right)\right) \]

   8. Simplified 98.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \left({\ell}^{\left(w + \color{blue}{1}\right)}\right)\right) \]

      2. pow-plus N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \left({\ell}^{w} \cdot \color{blue}{\ell}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{*.f64}\left(\left({\ell}^{w}\right), \color{blue}{\ell}\right)\right) \]

      4. pow-lowering-pow.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\ell, w\right), \ell\right)\right) \]

   10. Applied egg-rr 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot \left(\ell \cdot {\ell}^{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) \cdot {\ell}^{w}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1500.0) (exp (- 0.0 w)) (* l (* (- 1.0 w) (pow l w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = exp((0.0 - w));
	} else {
		tmp = l * ((1.0 - w) * pow(l, w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1500.0d0)) then
        tmp = exp((0.0d0 - w))
    else
        tmp = l * ((1.0d0 - w) * (l ** w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = l * ((1.0 - w) * Math.pow(l, w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1500.0:
		tmp = math.exp((0.0 - w))
	else:
		tmp = l * ((1.0 - w) * math.pow(l, w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1500.0)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64(l * Float64(Float64(1.0 - w) * (l ^ w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1500.0)
		tmp = exp((0.0 - w));
	else
		tmp = l * ((1.0 - w) * (l ^ w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1500.0], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(l * N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, w], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1500

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -1500 < w

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \color{blue}{\left(1 + w\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{w}\right)\right)\right) \]

   8. Simplified 98.0%

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

      1. *-commutative N/A

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

      2. unpow-prod-up N/A

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

      3. unpow1 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left({\ell}^{w} \cdot \left(1 - w\right)\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\left({\ell}^{w}\right), \color{blue}{\left(1 - w\right)}\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\ell, w\right), \left(\color{blue}{1} - w\right)\right)\right) \]

      8. --lowering--.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\ell, w\right), \mathsf{\_.f64}\left(1, \color{blue}{w}\right)\right)\right) \]

   10. Applied egg-rr 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\left(1 - w\right) \cdot {\ell}^{w}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(w + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1500.0) (exp (- 0.0 w)) (* (- 1.0 w) (pow l (+ w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = exp((0.0 - w));
	} else {
		tmp = (1.0 - w) * pow(l, (w + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1500.0d0)) then
        tmp = exp((0.0d0 - w))
    else
        tmp = (1.0d0 - w) * (l ** (w + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1500.0) {
		tmp = Math.exp((0.0 - w));
	} else {
		tmp = (1.0 - w) * Math.pow(l, (w + 1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1500.0:
		tmp = math.exp((0.0 - w))
	else:
		tmp = (1.0 - w) * math.pow(l, (w + 1.0))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1500.0)
		tmp = exp(Float64(0.0 - w));
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(w + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1500.0)
		tmp = exp((0.0 - w));
	else
		tmp = (1.0 - w) * (l ^ (w + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1500.0], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1500

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -1500 < w

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. --lowering--.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 98.3%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \color{blue}{\left(1 + w\right)}\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \mathsf{pow.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{w}\right)\right)\right) \]

   8. Simplified 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1500:\\ \;\;\;\;e^{0 - w}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(w + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in w around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
4. Step-by-step derivation

5. Simplified 97.0%

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

   1. *-commutative N/A

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

   2. exp-neg N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(e^{w}\right)}\right) \]

   5. exp-lowering-exp.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]
8. Add Preprocessing

Alternative 10: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\ell \cdot \frac{1 + w \cdot \mathsf{fma}\left(w, w, 0\right)}{1 + \left(\mathsf{fma}\left(w, w, 0\right) - w\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.7)
   (exp (- 0.0 w))
   (if (<= w 0.18)
     (* l (/ (+ 1.0 (* w (fma w w 0.0))) (+ 1.0 (- (fma w w 0.0) w))))
     0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.7) {
		tmp = exp((0.0 - w));
	} else if (w <= 0.18) {
		tmp = l * ((1.0 + (w * fma(w, w, 0.0))) / (1.0 + (fma(w, w, 0.0) - w)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.7)
		tmp = exp(Float64(0.0 - w));
	elseif (w <= 0.18)
		tmp = Float64(l * Float64(Float64(1.0 + Float64(w * fma(w, w, 0.0))) / Float64(1.0 + Float64(fma(w, w, 0.0) - w))));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.7], N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.18], N[(l * N[(N[(1.0 + N[(w * N[(w * w + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(w * w + 0.0), $MachinePrecision] - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.69999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in w around inf

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

      1. *-commutative N/A

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

      2. exp-to-pow N/A

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

      3. remove-double-neg N/A

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

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

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

      5. log-rec N/A

         \[\leadsto e^{\mathsf{neg}\left(\log \left(\frac{1}{\ell}\right) \cdot e^{w}\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      6. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \cdot e^{\mathsf{neg}\left(w\right)} \]

      7. mul-1-neg N/A

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

      8. +-rgt-identity N/A

         \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0} \cdot e^{\mathsf{neg}\left(\color{blue}{w}\right)} \]

      9. exp-sum N/A

         \[\leadsto e^{\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + 0\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      10. +-rgt-identity N/A

          \[\leadsto e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto e^{\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)} \]

      12. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(w\right)\right) + -1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) + \left(\mathsf{neg}\left(w\right)\right)\right)\right) \]

      14. unsub-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right) - w\right)\right) \]

      15. --lowering--.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)\right), w\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \ell, e^{w}, 0\right) - w}} \]

   6. Taylor expanded in w around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(-1 \cdot w\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(w\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - w\right)\right) \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, w\right)\right) \]

   8. Simplified 100.0%

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

   if -0.69999999999999996 < w < 0.17999999999999999

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 96.5%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \ell\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \ell\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \ell\right) \]

      3. --lowering--.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \ell\right) \]

   8. Simplified 96.5%

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

   10. Applied egg-rr 96.6%

       \[\leadsto \color{blue}{\frac{1 + w \cdot \mathsf{fma}\left(w, w, 0\right)}{1 + \left(\mathsf{fma}\left(w, w, 0\right) - w\right)}} \cdot \ell \]

   if 0.17999999999999999 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7:\\ \;\;\;\;e^{0 - w}\\ \mathbf{elif}\;w \leq 0.18:\\ \;\;\;\;\ell \cdot \frac{1 + w \cdot \mathsf{fma}\left(w, w, 0\right)}{1 + \left(\mathsf{fma}\left(w, w, 0\right) - w\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \ell \cdot e^{0 - w} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (exp (- 0.0 w))))

C

double code(double w, double l) {
	return l * exp((0.0 - w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * exp((0.0d0 - w))
end function

Java

public static double code(double w, double l) {
	return l * Math.exp((0.0 - w));
}

Python

def code(w, l):
	return l * math.exp((0.0 - w))

Julia

function code(w, l)
	return Float64(l * exp(Float64(0.0 - w)))
end

MATLAB

function tmp = code(w, l)
	tmp = l * exp((0.0 - w));
end

Wolfram

code[w_, l_] := N[(l * N[Exp[N[(0.0 - w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in w around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
4. Step-by-step derivation

5. Simplified 97.0%

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

6. Final simplification 97.0%

   \[\leadsto \ell \cdot e^{0 - w} \]
7. Add Preprocessing

Alternative 12: 93.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.1:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(w \cdot \left(w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right)\right)\right), \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), w \cdot w, -1\right)}, 0 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2e+103)
   (* l (* -0.16666666666666666 (* w (* w w))))
   (if (<= w 0.1)
     (*
      l
      (fma
       (fma
        (* (* w w) (* w (* w (fma w -0.16666666666666666 0.5))))
        (fma w -0.16666666666666666 0.5)
        -1.0)
       (/ 1.0 (fma (fma w -0.16666666666666666 0.5) (* w w) -1.0))
       (- 0.0 w)))
     0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2e+103) {
		tmp = l * (-0.16666666666666666 * (w * (w * w)));
	} else if (w <= 0.1) {
		tmp = l * fma(fma(((w * w) * (w * (w * fma(w, -0.16666666666666666, 0.5)))), fma(w, -0.16666666666666666, 0.5), -1.0), (1.0 / fma(fma(w, -0.16666666666666666, 0.5), (w * w), -1.0)), (0.0 - w));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2e+103)
		tmp = Float64(l * Float64(-0.16666666666666666 * Float64(w * Float64(w * w))));
	elseif (w <= 0.1)
		tmp = Float64(l * fma(fma(Float64(Float64(w * w) * Float64(w * Float64(w * fma(w, -0.16666666666666666, 0.5)))), fma(w, -0.16666666666666666, 0.5), -1.0), Float64(1.0 / fma(fma(w, -0.16666666666666666, 0.5), Float64(w * w), -1.0)), Float64(0.0 - w)));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2e+103], N[(l * N[(-0.16666666666666666 * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.1], N[(l * N[(N[(N[(N[(w * w), $MachinePrecision] * N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] * N[(w * w), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(0.0 - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -2e103

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(e^{w}\right)}\right) \]

      5. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right) \]

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in w around 0

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

   10. Simplified 90.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(\ell, 0.5, 0 - w \cdot \mathsf{fma}\left(\ell, 0.16666666666666666, 0\right)\right), 0\right) - \ell, \ell\right)} \]

   11. Taylor expanded in w around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left(\ell \cdot {w}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \ell\right) \cdot \color{blue}{{w}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(\ell \cdot \frac{-1}{6}\right) \cdot {\color{blue}{w}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\frac{-1}{6} \cdot {w}^{3}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{6} \cdot {w}^{3}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({w}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \left(w \cdot \color{blue}{\left(w \cdot w\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \left(w \cdot {w}^{\color{blue}{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\ell \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)} \]

   if -2e103 < w < 0.10000000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) + 1\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right), 1\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right), 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right), 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right), -1\right), 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(\frac{-1}{6} \cdot w + \frac{1}{2}\right), -1\right), 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(w \cdot \frac{-1}{6} + \frac{1}{2}\right), -1\right), 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 83.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   5. Simplified 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)} \cdot {\ell}^{\left(e^{w}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(w \cdot \left(w \cdot \frac{-1}{6} + \frac{1}{2}\right)\right) \cdot w + -1 \cdot w\right) + 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(w \cdot \left(w \cdot \frac{-1}{6} + \frac{1}{2}\right)\right) \cdot w + \left(\mathsf{neg}\left(w\right)\right)\right) + 1\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot \left(w \cdot \frac{-1}{6} + \frac{1}{2}\right)\right) \cdot w + \left(\left(\mathsf{neg}\left(w\right)\right) + 1\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot w\right) \cdot w + \left(\left(\mathsf{neg}\left(w\right)\right) + 1\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) + \left(\left(\mathsf{neg}\left(w\right)\right) + 1\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) + \left(1 + \left(\mathsf{neg}\left(w\right)\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) + \left(1 - w\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right), \left(w \cdot w\right), \left(1 - w\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), \left(w \cdot w\right), \left(1 - w\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), \mathsf{*.f64}\left(w, w\right), \left(1 - w\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      11. --lowering--.f64 83.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), \mathsf{*.f64}\left(w, w\right), \mathsf{\_.f64}\left(1, w\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   7. Applied egg-rr 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), w \cdot w, 1 - w\right)} \cdot {\ell}^{\left(e^{w}\right)} \]
   8. Step-by-step derivation

      1. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) + 1\right) - w\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) + 1\right) + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right)\right) - 1 \cdot 1}{\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) - 1} + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right)\right) - 1 \cdot 1\right) \cdot \frac{1}{\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) - 1} + \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\left(\left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right)\right) \cdot \left(\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right)\right) - 1 \cdot 1\right), \left(\frac{1}{\left(w \cdot \frac{-1}{6} + \frac{1}{2}\right) \cdot \left(w \cdot w\right) - 1}\right), \left(\mathsf{neg}\left(w\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\ell}, \mathsf{exp.f64}\left(w\right)\right)\right) \]

   9. Applied egg-rr 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(w \cdot \left(w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right)\right)\right) \cdot \left(w \cdot w\right), \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), w \cdot w, -1\right)}, 0 - w\right)} \cdot {\ell}^{\left(e^{w}\right)} \]

   10. Taylor expanded in w around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right)\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), -1\right), \mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), \mathsf{*.f64}\left(w, w\right), -1\right)\right), \mathsf{\_.f64}\left(0, w\right)\right), \color{blue}{\ell}\right) \]
   11. Step-by-step derivation

   12. Simplified 89.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(w \cdot \left(w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right)\right)\right) \cdot \left(w \cdot w\right), \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), w \cdot w, -1\right)}, 0 - w\right) \cdot \color{blue}{\ell} \]

   if 0.10000000000000001 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+103}:\\ \;\;\;\;\ell \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.1:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(w \cdot \left(w \cdot \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right)\right)\right), \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), w \cdot w, -1\right)}, 0 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.115)
   (* l (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0))
   0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.115) {
		tmp = l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.115)
		tmp = Float64(l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.115], N[(l * N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.115000000000000005

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 97.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)}, \ell\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) + 1\right), \ell\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right), 1\right), \ell\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)\right), 1\right), \ell\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + -1\right), 1\right), \ell\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right), -1\right), 1\right), \ell\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(\frac{-1}{6} \cdot w + \frac{1}{2}\right), -1\right), 1\right), \ell\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \left(w \cdot \frac{-1}{6} + \frac{1}{2}\right), -1\right), 1\right), \ell\right) \]

      8. accelerator-lowering-fma.f64 87.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \frac{-1}{6}, \frac{1}{2}\right), -1\right), 1\right), \ell\right) \]

   8. Simplified 87.5%

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

   if 0.115000000000000005 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.115:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.023:\\ \;\;\;\;\ell \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.13:\\ \;\;\;\;\ell \cdot \left(w + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.023)
   (* l (* -0.16666666666666666 (* w (* w w))))
   (if (<= w 0.13) (* l (+ w 1.0)) 0.0)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.023) {
		tmp = l * (-0.16666666666666666 * (w * (w * w)));
	} else if (w <= 0.13) {
		tmp = l * (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.023d0)) then
        tmp = l * ((-0.16666666666666666d0) * (w * (w * w)))
    else if (w <= 0.13d0) then
        tmp = l * (w + 1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.023) {
		tmp = l * (-0.16666666666666666 * (w * (w * w)));
	} else if (w <= 0.13) {
		tmp = l * (w + 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.023:
		tmp = l * (-0.16666666666666666 * (w * (w * w)))
	elif w <= 0.13:
		tmp = l * (w + 1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.023)
		tmp = Float64(l * Float64(-0.16666666666666666 * Float64(w * Float64(w * w))));
	elseif (w <= 0.13)
		tmp = Float64(l * Float64(w + 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.023)
		tmp = l * (-0.16666666666666666 * (w * (w * w)));
	elseif (w <= 0.13)
		tmp = l * (w + 1.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.023], N[(l * N[(-0.16666666666666666 * N[(w * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 0.13], N[(l * N[(w + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if w < -0.023

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. exp-neg N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\ell, \color{blue}{\left(e^{w}\right)}\right) \]

      5. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\ell, \mathsf{exp.f64}\left(w\right)\right) \]

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in w around 0

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

   10. Simplified 65.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(\ell, 0.5, 0 - w \cdot \mathsf{fma}\left(\ell, 0.16666666666666666, 0\right)\right), 0\right) - \ell, \ell\right)} \]

   11. Taylor expanded in w around inf

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left(\ell \cdot {w}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot \ell\right) \cdot \color{blue}{{w}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(\ell \cdot \frac{-1}{6}\right) \cdot {\color{blue}{w}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\frac{-1}{6} \cdot {w}^{3}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{6} \cdot {w}^{3}\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({w}^{3}\right)}\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \left(w \cdot \color{blue}{\left(w \cdot w\right)}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \left(w \cdot {w}^{\color{blue}{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, \color{blue}{\left({w}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, \left(w \cdot \color{blue}{w}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 71.4%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \color{blue}{w}\right)\right)\right)\right) \]

   13. Simplified 71.4%

       \[\leadsto \color{blue}{\ell \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)} \]

   if -0.023 < w < 0.13

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 96.5%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \ell\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \ell\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \ell\right) \]

      3. --lowering--.f64 96.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \ell\right) \]

   8. Simplified 96.5%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \ell\right) \]

      2. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(0 - w\right)\right), \ell\right) \]

      3. flip3-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{0}^{3} - {w}^{3}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{0 - {w}^{3}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{\mathsf{neg}\left({w}^{3}\right)}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      6. cube-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(\mathsf{neg}\left(w\right)\right)}^{3}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      7. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(0 - w\right)}^{3}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      8. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(0 - w\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - w\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(\left(0 - w\right) \cdot \left(0 - w\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(\left(\mathsf{neg}\left(w\right)\right) \cdot \left(0 - w\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      11. sub0-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(\left(\mathsf{neg}\left(w\right)\right) \cdot \left(\mathsf{neg}\left(w\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      12. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{\left(w \cdot w\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      13. pow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{\left(\frac{3}{2}\right)} \cdot {w}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      14. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{0 \cdot 0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{0 + \left(w \cdot w + 0 \cdot w\right)}\right), \ell\right) \]

      16. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{w \cdot w + 0 \cdot w}\right), \ell\right) \]

      17. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{w \cdot \left(w + 0\right)}\right), \ell\right) \]

      18. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{w \cdot \left(0 + w\right)}\right), \ell\right) \]

      19. +-lft-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{w \cdot w}\right), \ell\right) \]

      20. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + \frac{{w}^{3}}{{w}^{2}}\right), \ell\right) \]

      21. pow-div N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + {w}^{\left(3 - 2\right)}\right), \ell\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + {w}^{1}\right), \ell\right) \]

      23. unpow1 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 + w\right), \ell\right) \]

   10. Applied egg-rr 96.6%

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

   if 0.13 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.023:\\ \;\;\;\;\ell \cdot \left(-0.16666666666666666 \cdot \left(w \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{elif}\;w \leq 0.13:\\ \;\;\;\;\ell \cdot \left(w + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.085:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.085) (* l (fma w (fma w 0.5 -1.0) 1.0)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.085) {
		tmp = l * fma(w, fma(w, 0.5, -1.0), 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.085)
		tmp = Float64(l * fma(w, fma(w, 0.5, -1.0), 1.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.085], N[(l * N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.0850000000000000061

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 97.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)}, \ell\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(w \cdot \left(\frac{1}{2} \cdot w - 1\right) + 1\right), \ell\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(\frac{1}{2} \cdot w - 1\right), 1\right), \ell\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(\frac{1}{2} \cdot w + \left(\mathsf{neg}\left(1\right)\right)\right), 1\right), \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \frac{1}{2} + \left(\mathsf{neg}\left(1\right)\right)\right), 1\right), \ell\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \left(w \cdot \frac{1}{2} + -1\right), 1\right), \ell\right) \]

      6. accelerator-lowering-fma.f64 84.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fma.f64}\left(w, \mathsf{fma.f64}\left(w, \frac{1}{2}, -1\right), 1\right), \ell\right) \]

   8. Simplified 84.3%

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

   if 0.0850000000000000061 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.085:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.7% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.075:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.075) (* l (- 1.0 w)) 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.075) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.075d0) then
        tmp = l * (1.0d0 - w)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.075) {
		tmp = l * (1.0 - w);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.075:
		tmp = l * (1.0 - w)
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.075)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.075)
		tmp = l * (1.0 - w);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.075], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.0749999999999999972

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{neg.f64}\left(w\right)\right), \color{blue}{\ell}\right) \]
   4. Step-by-step derivation

   5. Simplified 97.8%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + -1 \cdot w\right)}, \ell\right) \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 + \left(\mathsf{neg}\left(w\right)\right)\right), \ell\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(1 - w\right), \ell\right) \]

      3. --lowering--.f64 74.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, w\right), \ell\right) \]

   8. Simplified 74.1%

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

   if 0.0749999999999999972 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 0.075:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.8% accurate, 44.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.14:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w 0.14) l 0.0))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.14) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 0.14d0) then
        tmp = l
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.14) {
		tmp = l;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.14:
		tmp = l
	else:
		tmp = 0.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.14)
		tmp = l;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.14)
		tmp = l;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.14], l, 0.0]

Derivation

1. Split input into 2 regimes

2. if w < 0.14000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell} \]
   4. Step-by-step derivation

   5. Simplified 63.2%

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

   if 0.14000000000000001 < w

   1. Initial program 100.0%

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

      1. exp-neg N/A

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

      2. sqr-pow N/A

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

      3. pow-prod-up N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      11. mul0-lft N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

      13. +-inverses N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

      15. flip-- N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

      16. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

      18. associate-/r/ N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

          \[\leadsto \frac{1}{e^{w} \cdot 1} \]

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 16.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 0.0)

C

double code(double w, double l) {
	return 0.0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = 0.0d0
end function

Java

public static double code(double w, double l) {
	return 0.0;
}

Python

def code(w, l):
	return 0.0

Julia

function code(w, l)
	return 0.0
end

MATLAB

function tmp = code(w, l)
	tmp = 0.0;
end

Wolfram

code[w_, l_] := 0.0

Derivation

1. Initial program 99.6%

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

   1. exp-neg N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-up N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 - 0}{\color{blue}{\frac{e^{w}}{2}} - \frac{e^{w}}{2}}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{\color{blue}{e^{w}}}{2} - \frac{e^{w}}{2}}\right)} \]

   8. mul0-lft N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - w \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot w}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   11. mul0-lft N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{\frac{e^{w}}{\color{blue}{2}} - \frac{e^{w}}{2}}\right)} \]

   13. +-inverses N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0}\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(\frac{0 \cdot 0 - 0 \cdot 0}{0 + \color{blue}{0}}\right)} \]

   15. flip-- N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{\left(0 - \color{blue}{0}\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot {\ell}^{0} \]

   17. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w}} \cdot 1 \]

   18. associate-/r/ N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

       \[\leadsto \frac{1}{e^{w} \cdot 1} \]

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

4. Applied egg-rr 19.0%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))