Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.3% → 98.6%

Time: 11.2s

Alternatives: 8

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -4.4769121955640854e+57
  2. x = 3.6521367420042234e+73

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t_0 - x}{e^{wj} + t_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t_0}{e^{wj} + t_0} \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x + x \cdot \left(wj \cdot -2\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 2.2e-13)
     (- (fma wj wj (+ x (* x (* wj -2.0)))) (pow wj 3.0))
     (+ wj (/ (- (/ x (exp wj)) wj) (+ wj 1.0))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 2.2e-13) {
		tmp = fma(wj, wj, (x + (x * (wj * -2.0)))) - pow(wj, 3.0);
	} else {
		tmp = wj + (((x / exp(wj)) - wj) / (wj + 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 2.2e-13)
		tmp = Float64(fma(wj, wj, Float64(x + Float64(x * Float64(wj * -2.0)))) - (wj ^ 3.0));
	else
		tmp = Float64(wj + Float64(Float64(Float64(x / exp(wj)) - wj) / Float64(wj + 1.0)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.2e-13], N[(N[(wj * wj + N[(x + N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision], N[(wj + N[(N[(N[(x / N[Exp[wj], $MachinePrecision]), $MachinePrecision] - wj), $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 2.19999999999999997e-13

   1. Initial program 78.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 78.1%

         \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

      2. associate-/l* 78.1%

         \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      3. distribute-rgt1-in 78.1%

         \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      4. associate-/l* 78.1%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      5. *-inverses 78.1%

         \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      6. /-rgt-identity 78.1%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      7. distribute-rgt1-in 78.6%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

      8. associate-/l/ 78.6%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

      9. div-sub 78.6%

         \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   4. Taylor expanded in wj around 0 98.8%

      \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 98.8%

         \[\leadsto \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

      2. +-commutative 98.8%

         \[\leadsto \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right)} + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      3. fma-def 98.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2, wj \cdot x, x\right)} + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      4. *-commutative 98.8%

         \[\leadsto \mathsf{fma}\left(-2, \color{blue}{x \cdot wj}, x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

      5. +-commutative 98.8%

         \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + -1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right)} \]

      6. mul-1-neg 98.8%

         \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + \color{blue}{\left(-{wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)}\right) \]

      7. unsub-neg 98.8%

         \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - {wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)} \]

   6. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) - {wj}^{3} \cdot \left(1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right)\right)\right)} \]

   7. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) - \color{blue}{{wj}^{3}}\right) \]

   8. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right) + {wj}^{2}\right) - {wj}^{3}} \]
   9. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\left({wj}^{2} + x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right)\right)} - {wj}^{3} \]

      2. unpow2 99.4%

         \[\leadsto \left(\color{blue}{wj \cdot wj} + x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right)\right) - {wj}^{3} \]

      3. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right)\right)} - {wj}^{3} \]

      4. distribute-lft-in 99.4%

         \[\leadsto \mathsf{fma}\left(wj, wj, \color{blue}{x \cdot 1 + x \cdot \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)}\right) - {wj}^{3} \]

      5. *-rgt-identity 99.4%

         \[\leadsto \mathsf{fma}\left(wj, wj, \color{blue}{x} + x \cdot \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right) - {wj}^{3} \]

      6. +-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \color{blue}{\left(2.5 \cdot {wj}^{2} + -2 \cdot wj\right)}\right) - {wj}^{3} \]

      7. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \left(\color{blue}{{wj}^{2} \cdot 2.5} + -2 \cdot wj\right)\right) - {wj}^{3} \]

      8. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \color{blue}{\mathsf{fma}\left({wj}^{2}, 2.5, -2 \cdot wj\right)}\right) - {wj}^{3} \]

      9. unpow2 99.4%

         \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \mathsf{fma}\left(\color{blue}{wj \cdot wj}, 2.5, -2 \cdot wj\right)\right) - {wj}^{3} \]

      10. *-commutative 99.4%

          \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \mathsf{fma}\left(wj \cdot wj, 2.5, \color{blue}{wj \cdot -2}\right)\right) - {wj}^{3} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x + x \cdot \mathsf{fma}\left(wj \cdot wj, 2.5, wj \cdot -2\right)\right) - {wj}^{3}} \]

   11. Taylor expanded in wj around 0 99.4%

       \[\leadsto \mathsf{fma}\left(wj, wj, x + \color{blue}{-2 \cdot \left(wj \cdot x\right)}\right) - {wj}^{3} \]
   12. Step-by-step derivation

       1. *-commutative 99.4%

          \[\leadsto \mathsf{fma}\left(wj, wj, x + \color{blue}{\left(wj \cdot x\right) \cdot -2}\right) - {wj}^{3} \]

       2. *-commutative 99.4%

          \[\leadsto \mathsf{fma}\left(wj, wj, x + \color{blue}{\left(x \cdot wj\right)} \cdot -2\right) - {wj}^{3} \]

       3. associate-*r* 99.4%

          \[\leadsto \mathsf{fma}\left(wj, wj, x + \color{blue}{x \cdot \left(wj \cdot -2\right)}\right) - {wj}^{3} \]

       4. *-commutative 99.4%

          \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \color{blue}{\left(-2 \cdot wj\right)}\right) - {wj}^{3} \]

   13. Simplified 99.4%

       \[\leadsto \mathsf{fma}\left(wj, wj, x + \color{blue}{x \cdot \left(-2 \cdot wj\right)}\right) - {wj}^{3} \]

   if 2.19999999999999997e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.2%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 95.2%

         \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

      2. associate-/l* 95.2%

         \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      3. distribute-rgt1-in 95.2%

         \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      4. associate-/l* 95.2%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      5. *-inverses 96.7%

         \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      6. /-rgt-identity 96.7%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      7. distribute-rgt1-in 99.7%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

      8. associate-/l/ 99.7%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

      9. div-sub 99.7%

         \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 2.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x + x \cdot \left(wj \cdot -2\right)\right) - {wj}^{3}\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{\frac{x}{e^{wj}} - wj}{wj + 1}\\ \end{array} \]

Alternative 2: 96.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-2, wj \cdot x, x\right) + \left(wj \cdot wj - {wj}^{3}\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+ (fma -2.0 (* wj x) x) (- (* wj wj) (pow wj 3.0))))

C

double code(double wj, double x) {
	return fma(-2.0, (wj * x), x) + ((wj * wj) - pow(wj, 3.0));
}

Julia

function code(wj, x)
	return Float64(fma(-2.0, Float64(wj * x), x) + Float64(Float64(wj * wj) - (wj ^ 3.0)))
end

Wolfram

code[wj_, x_] := N[(N[(-2.0 * N[(wj * x), $MachinePrecision] + x), $MachinePrecision] + N[(N[(wj * wj), $MachinePrecision] - N[Power[wj, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 82.6%

      \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   2. associate-/l* 82.6%

      \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   3. distribute-rgt1-in 82.6%

      \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   4. associate-/l* 82.6%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   5. *-inverses 83.0%

      \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   6. /-rgt-identity 83.0%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

   8. associate-/l/ 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

   9. div-sub 84.1%

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Simplified 84.1%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

4. Taylor expanded in wj around 0 97.5%

   \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 97.5%

      \[\leadsto \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   2. +-commutative 97.5%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right)} + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

   3. fma-def 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, wj \cdot x, x\right)} + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

   4. *-commutative 97.5%

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{x \cdot wj}, x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

   5. +-commutative 97.5%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + -1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right)} \]

   6. mul-1-neg 97.5%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + \color{blue}{\left(-{wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)}\right) \]

   7. unsub-neg 97.5%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - {wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)} \]

6. Simplified 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) - {wj}^{3} \cdot \left(1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right)\right)\right)} \]

7. Taylor expanded in x around 0 97.6%

   \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) - \color{blue}{{wj}^{3}}\right) \]

8. Taylor expanded in x around 0 97.9%

   \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} - {wj}^{3}\right)} \]
9. Step-by-step derivation

   1. unpow2 97.9%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\color{blue}{wj \cdot wj} - {wj}^{3}\right) \]

10. Simplified 97.9%

    \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left(wj \cdot wj - {wj}^{3}\right)} \]

11. Final simplification 97.9%

    \[\leadsto \mathsf{fma}\left(-2, wj \cdot x, x\right) + \left(wj \cdot wj - {wj}^{3}\right) \]

Alternative 3: 95.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, wj, \mathsf{fma}\left(x, wj \cdot \left(-2 + wj \cdot 2.5\right), x\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma wj wj (fma x (* wj (+ -2.0 (* wj 2.5))) x)))

C

double code(double wj, double x) {
	return fma(wj, wj, fma(x, (wj * (-2.0 + (wj * 2.5))), x));
}

Julia

function code(wj, x)
	return fma(wj, wj, fma(x, Float64(wj * Float64(-2.0 + Float64(wj * 2.5))), x))
end

Wolfram

code[wj_, x_] := N[(wj * wj + N[(x * N[(wj * N[(-2.0 + N[(wj * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 82.6%

      \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   2. associate-/l* 82.6%

      \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   3. distribute-rgt1-in 82.6%

      \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   4. associate-/l* 82.6%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   5. *-inverses 83.0%

      \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   6. /-rgt-identity 83.0%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

   8. associate-/l/ 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

   9. div-sub 84.1%

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Simplified 84.1%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

4. Taylor expanded in wj around 0 97.5%

   \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-+r+ 97.5%

      \[\leadsto \color{blue}{\left(x + -2 \cdot \left(wj \cdot x\right)\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]

   2. +-commutative 97.5%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + x\right)} + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

   3. fma-def 97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, wj \cdot x, x\right)} + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

   4. *-commutative 97.5%

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{x \cdot wj}, x\right) + \left(-1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right) \]

   5. +-commutative 97.5%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + -1 \cdot \left({wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)\right)} \]

   6. mul-1-neg 97.5%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + \color{blue}{\left(-{wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)}\right) \]

   7. unsub-neg 97.5%

      \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) - {wj}^{3} \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + 1.5 \cdot x\right) + 0.6666666666666666 \cdot x\right)\right)\right)\right)} \]

6. Simplified 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) - {wj}^{3} \cdot \left(1 + \mathsf{fma}\left(x, -3, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 5\right)\right)\right)\right)} \]

7. Taylor expanded in x around 0 97.6%

   \[\leadsto \mathsf{fma}\left(-2, x \cdot wj, x\right) + \left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) - \color{blue}{{wj}^{3}}\right) \]

8. Taylor expanded in x around 0 97.9%

   \[\leadsto \color{blue}{\left(x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right) + {wj}^{2}\right) - {wj}^{3}} \]
9. Step-by-step derivation

   1. +-commutative 97.9%

      \[\leadsto \color{blue}{\left({wj}^{2} + x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right)\right)} - {wj}^{3} \]

   2. unpow2 97.9%

      \[\leadsto \left(\color{blue}{wj \cdot wj} + x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right)\right) - {wj}^{3} \]

   3. fma-def 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x \cdot \left(1 + \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right)\right)} - {wj}^{3} \]

   4. distribute-lft-in 97.9%

      \[\leadsto \mathsf{fma}\left(wj, wj, \color{blue}{x \cdot 1 + x \cdot \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)}\right) - {wj}^{3} \]

   5. *-rgt-identity 97.9%

      \[\leadsto \mathsf{fma}\left(wj, wj, \color{blue}{x} + x \cdot \left(-2 \cdot wj + 2.5 \cdot {wj}^{2}\right)\right) - {wj}^{3} \]

   6. +-commutative 97.9%

      \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \color{blue}{\left(2.5 \cdot {wj}^{2} + -2 \cdot wj\right)}\right) - {wj}^{3} \]

   7. *-commutative 97.9%

      \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \left(\color{blue}{{wj}^{2} \cdot 2.5} + -2 \cdot wj\right)\right) - {wj}^{3} \]

   8. fma-def 97.9%

      \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \color{blue}{\mathsf{fma}\left({wj}^{2}, 2.5, -2 \cdot wj\right)}\right) - {wj}^{3} \]

   9. unpow2 97.9%

      \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \mathsf{fma}\left(\color{blue}{wj \cdot wj}, 2.5, -2 \cdot wj\right)\right) - {wj}^{3} \]

   10. *-commutative 97.9%

       \[\leadsto \mathsf{fma}\left(wj, wj, x + x \cdot \mathsf{fma}\left(wj \cdot wj, 2.5, \color{blue}{wj \cdot -2}\right)\right) - {wj}^{3} \]

10. Simplified 97.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x + x \cdot \mathsf{fma}\left(wj \cdot wj, 2.5, wj \cdot -2\right)\right) - {wj}^{3}} \]

11. Taylor expanded in wj around 0 97.2%

    \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 + 2.5 \cdot x\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative 97.2%

       \[\leadsto \color{blue}{\left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 + 2.5 \cdot x\right)\right) + x} \]

13. Simplified 97.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, \mathsf{fma}\left(x, wj \cdot \left(-2 + wj \cdot 2.5\right), x\right)\right)} \]

14. Final simplification 97.5%

    \[\leadsto \mathsf{fma}\left(wj, wj, \mathsf{fma}\left(x, wj \cdot \left(-2 + wj \cdot 2.5\right), x\right)\right) \]

Alternative 4: 83.1% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj - wj \cdot \left(wj \cdot wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 7.2e-24) x (- (* wj wj) (* wj (* wj wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-24) {
		tmp = x;
	} else {
		tmp = (wj * wj) - (wj * (wj * wj));
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 7.2d-24) then
        tmp = x
    else
        tmp = (wj * wj) - (wj * (wj * wj))
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 7.2e-24) {
		tmp = x;
	} else {
		tmp = (wj * wj) - (wj * (wj * wj));
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 7.2e-24:
		tmp = x
	else:
		tmp = (wj * wj) - (wj * (wj * wj))
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 7.2e-24)
		tmp = x;
	else
		tmp = Float64(Float64(wj * wj) - Float64(wj * Float64(wj * wj)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 7.2e-24)
		tmp = x;
	else
		tmp = (wj * wj) - (wj * (wj * wj));
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 7.2e-24], x, N[(N[(wj * wj), $MachinePrecision] - N[(wj * N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.2000000000000002e-24

   1. Initial program 84.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 84.1%

         \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

      2. associate-/l* 84.1%

         \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      3. distribute-rgt1-in 84.1%

         \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      4. associate-/l* 84.1%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      5. *-inverses 84.1%

         \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      6. /-rgt-identity 84.1%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      7. distribute-rgt1-in 85.3%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

      8. associate-/l/ 85.3%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

      9. div-sub 85.3%

         \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   4. Taylor expanded in wj around 0 90.7%

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

   if 7.2000000000000002e-24 < wj

   1. Initial program 45.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Step-by-step derivation

      1. div-sub 45.3%

         \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

      2. associate-/l* 45.2%

         \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      3. distribute-rgt1-in 45.2%

         \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      4. associate-/l* 45.2%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      5. *-inverses 55.2%

         \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      6. /-rgt-identity 55.2%

         \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

      7. distribute-rgt1-in 55.2%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

      8. associate-/l/ 55.4%

         \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

      9. div-sub 55.4%

         \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   3. Simplified 55.4%

      \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

   4. Taylor expanded in x around 0 45.3%

      \[\leadsto \color{blue}{wj - \frac{wj}{1 + wj}} \]
   5. Step-by-step derivation

      1. +-commutative 45.3%

         \[\leadsto wj - \frac{wj}{\color{blue}{wj + 1}} \]

   6. Simplified 45.3%

      \[\leadsto \color{blue}{wj - \frac{wj}{wj + 1}} \]

   7. Taylor expanded in wj around 0 58.6%

      \[\leadsto \color{blue}{-1 \cdot {wj}^{3} + {wj}^{2}} \]
   8. Step-by-step derivation

      1. +-commutative 58.6%

         \[\leadsto \color{blue}{{wj}^{2} + -1 \cdot {wj}^{3}} \]

      2. mul-1-neg 58.6%

         \[\leadsto {wj}^{2} + \color{blue}{\left(-{wj}^{3}\right)} \]

      3. unsub-neg 58.6%

         \[\leadsto \color{blue}{{wj}^{2} - {wj}^{3}} \]

      4. unpow2 58.6%

         \[\leadsto \color{blue}{wj \cdot wj} - {wj}^{3} \]

   9. Simplified 58.6%

      \[\leadsto \color{blue}{wj \cdot wj - {wj}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 58.6%

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

       2. cancel-sign-sub-inv 58.6%

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

   11. Applied egg-rr 58.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 7.2 \cdot 10^{-24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj - wj \cdot \left(wj \cdot wj\right)\\ \end{array} \]

Alternative 5: 95.7% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ x + \left(x \cdot \left(wj \cdot -2\right) + wj \cdot wj\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (+ (* x (* wj -2.0)) (* wj wj))))

C

double code(double wj, double x) {
	return x + ((x * (wj * -2.0)) + (wj * wj));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((x * (wj * (-2.0d0))) + (wj * wj))
end function

Java

public static double code(double wj, double x) {
	return x + ((x * (wj * -2.0)) + (wj * wj));
}

Python

def code(wj, x):
	return x + ((x * (wj * -2.0)) + (wj * wj))

Julia

function code(wj, x)
	return Float64(x + Float64(Float64(x * Float64(wj * -2.0)) + Float64(wj * wj)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + ((x * (wj * -2.0)) + (wj * wj));
end

Wolfram

code[wj_, x_] := N[(x + N[(N[(x * N[(wj * -2.0), $MachinePrecision]), $MachinePrecision] + N[(wj * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 82.6%

      \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   2. associate-/l* 82.6%

      \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   3. distribute-rgt1-in 82.6%

      \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   4. associate-/l* 82.6%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   5. *-inverses 83.0%

      \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   6. /-rgt-identity 83.0%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

   8. associate-/l/ 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

   9. div-sub 84.1%

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Simplified 84.1%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

4. Taylor expanded in wj around 0 97.2%

   \[\leadsto \color{blue}{x + \left(-2 \cdot \left(wj \cdot x\right) + {wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 97.2%

      \[\leadsto x + \color{blue}{\left({wj}^{2} \cdot \left(1 - \left(-4 \cdot x + 1.5 \cdot x\right)\right) + -2 \cdot \left(wj \cdot x\right)\right)} \]

   2. fma-def 97.2%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left({wj}^{2}, 1 - \left(-4 \cdot x + 1.5 \cdot x\right), -2 \cdot \left(wj \cdot x\right)\right)} \]

   3. unpow2 97.2%

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{wj \cdot wj}, 1 - \left(-4 \cdot x + 1.5 \cdot x\right), -2 \cdot \left(wj \cdot x\right)\right) \]

   4. sub-neg 97.2%

      \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, \color{blue}{1 + \left(-\left(-4 \cdot x + 1.5 \cdot x\right)\right)}, -2 \cdot \left(wj \cdot x\right)\right) \]

   5. distribute-rgt-out 97.2%

      \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \left(-\color{blue}{x \cdot \left(-4 + 1.5\right)}\right), -2 \cdot \left(wj \cdot x\right)\right) \]

   6. distribute-rgt-neg-in 97.2%

      \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + \color{blue}{x \cdot \left(-\left(-4 + 1.5\right)\right)}, -2 \cdot \left(wj \cdot x\right)\right) \]

   7. metadata-eval 97.2%

      \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot \left(-\color{blue}{-2.5}\right), -2 \cdot \left(wj \cdot x\right)\right) \]

   8. metadata-eval 97.2%

      \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot \color{blue}{2.5}, -2 \cdot \left(wj \cdot x\right)\right) \]

   9. associate-*r* 97.2%

      \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, \color{blue}{\left(-2 \cdot wj\right) \cdot x}\right) \]

   10. *-commutative 97.2%

       \[\leadsto x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, \color{blue}{x \cdot \left(-2 \cdot wj\right)}\right) \]

6. Simplified 97.2%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(wj \cdot wj, 1 + x \cdot 2.5, x \cdot \left(-2 \cdot wj\right)\right)} \]
7. Step-by-step derivation

   1. fma-udef 97.2%

      \[\leadsto x + \color{blue}{\left(\left(wj \cdot wj\right) \cdot \left(1 + x \cdot 2.5\right) + x \cdot \left(-2 \cdot wj\right)\right)} \]

   2. associate-*l* 97.2%

      \[\leadsto x + \left(\color{blue}{wj \cdot \left(wj \cdot \left(1 + x \cdot 2.5\right)\right)} + x \cdot \left(-2 \cdot wj\right)\right) \]

   3. *-commutative 97.2%

      \[\leadsto x + \left(wj \cdot \left(wj \cdot \left(1 + x \cdot 2.5\right)\right) + x \cdot \color{blue}{\left(wj \cdot -2\right)}\right) \]

8. Applied egg-rr 97.2%

   \[\leadsto x + \color{blue}{\left(wj \cdot \left(wj \cdot \left(1 + x \cdot 2.5\right)\right) + x \cdot \left(wj \cdot -2\right)\right)} \]

9. Taylor expanded in x around 0 97.5%

   \[\leadsto x + \left(wj \cdot \color{blue}{wj} + x \cdot \left(wj \cdot -2\right)\right) \]

10. Final simplification 97.5%

    \[\leadsto x + \left(x \cdot \left(wj \cdot -2\right) + wj \cdot wj\right) \]

Alternative 6: 83.9% accurate, 44.7× speedup

Math

\[\begin{array}{l} \\ x + -2 \cdot \left(wj \cdot x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ x (* -2.0 (* wj x))))

C

double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + ((-2.0d0) * (wj * x))
end function

Java

public static double code(double wj, double x) {
	return x + (-2.0 * (wj * x));
}

Python

def code(wj, x):
	return x + (-2.0 * (wj * x))

Julia

function code(wj, x)
	return Float64(x + Float64(-2.0 * Float64(wj * x)))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (-2.0 * (wj * x));
end

Wolfram

code[wj_, x_] := N[(x + N[(-2.0 * N[(wj * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 82.6%

      \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   2. associate-/l* 82.6%

      \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   3. distribute-rgt1-in 82.6%

      \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   4. associate-/l* 82.6%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   5. *-inverses 83.0%

      \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   6. /-rgt-identity 83.0%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

   8. associate-/l/ 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

   9. div-sub 84.1%

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Simplified 84.1%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

4. Taylor expanded in wj around 0 87.7%

   \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
5. Step-by-step derivation

   1. *-commutative 87.7%

      \[\leadsto x + -2 \cdot \color{blue}{\left(x \cdot wj\right)} \]

6. Simplified 87.7%

   \[\leadsto \color{blue}{x + -2 \cdot \left(x \cdot wj\right)} \]

7. Final simplification 87.7%

   \[\leadsto x + -2 \cdot \left(wj \cdot x\right) \]

Alternative 7: 4.4% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 wj)

C

double code(double wj, double x) {
	return wj;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj
end function

Java

public static double code(double wj, double x) {
	return wj;
}

Python

def code(wj, x):
	return wj

Julia

function code(wj, x)
	return wj
end

MATLAB

function tmp = code(wj, x)
	tmp = wj;
end

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 82.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 82.6%

      \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   2. associate-/l* 82.6%

      \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   3. distribute-rgt1-in 82.6%

      \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   4. associate-/l* 82.6%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   5. *-inverses 83.0%

      \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   6. /-rgt-identity 83.0%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

   8. associate-/l/ 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

   9. div-sub 84.1%

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Simplified 84.1%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

4. Taylor expanded in wj around inf 4.3%

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

5. Final simplification 4.3%

   \[\leadsto wj \]

Alternative 8: 83.3% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 x)

C

double code(double wj, double x) {
	return x;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double wj, double x) {
	return x;
}

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

function tmp = code(wj, x)
	tmp = x;
end

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 82.6%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Step-by-step derivation

   1. div-sub 82.6%

      \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)} \]

   2. associate-/l* 82.6%

      \[\leadsto wj - \left(\color{blue}{\frac{wj}{\frac{e^{wj} + wj \cdot e^{wj}}{e^{wj}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   3. distribute-rgt1-in 82.6%

      \[\leadsto wj - \left(\frac{wj}{\frac{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}{e^{wj}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   4. associate-/l* 82.6%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{\frac{wj + 1}{\frac{e^{wj}}{e^{wj}}}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   5. *-inverses 83.0%

      \[\leadsto wj - \left(\frac{wj}{\frac{wj + 1}{\color{blue}{1}}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   6. /-rgt-identity 83.0%

      \[\leadsto wj - \left(\frac{wj}{\color{blue}{wj + 1}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \]

   7. distribute-rgt1-in 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right) \]

   8. associate-/l/ 84.1%

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{e^{wj}}}{wj + 1}}\right) \]

   9. div-sub 84.1%

      \[\leadsto wj - \color{blue}{\frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

3. Simplified 84.1%

   \[\leadsto \color{blue}{wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}} \]

4. Taylor expanded in wj around 0 87.6%

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

5. Final simplification 87.6%

   \[\leadsto x \]

Developer target: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj)))))))

C

double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = wj - ((wj / (wj + 1.0d0)) - (x / (exp(wj) + (wj * exp(wj)))))
end function

Java

public static double code(double wj, double x) {
	return wj - ((wj / (wj + 1.0)) - (x / (Math.exp(wj) + (wj * Math.exp(wj)))));
}

Python

def code(wj, x):
	return wj - ((wj / (wj + 1.0)) - (x / (math.exp(wj) + (wj * math.exp(wj)))))

Julia

function code(wj, x)
	return Float64(wj - Float64(Float64(wj / Float64(wj + 1.0)) - Float64(x / Float64(exp(wj) + Float64(wj * exp(wj))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = wj - ((wj / (wj + 1.0)) - (x / (exp(wj) + (wj * exp(wj)))));
end

Wolfram

code[wj_, x_] := N[(wj - N[(N[(wj / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[Exp[wj], $MachinePrecision] + N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023275 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))