Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.1% → 98.9%

Time: 11.6s

Alternatives: 11

Speedup: 331.0×

• could not determine a ground truth

  1. wj = 3.962655850681198e+218
  2. x = -1.3444548657631587e+286

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: 77.1% 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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{e^{wj} \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-16)
     (fma
      wj
      (fma
       wj
       (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
       (* x -2.0))
      x)
     (- wj (fma wj (/ 1.0 (+ wj 1.0)) (/ x (* (exp wj) (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-16) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = wj - fma(wj, (1.0 / (wj + 1.0)), (x / (exp(wj) * (-1.0 - wj))));
	}
	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))) <= 5e-16)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - fma(wj, Float64(1.0 / Float64(wj + 1.0)), Float64(x / Float64(exp(wj) * Float64(-1.0 - wj)))));
	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], 5e-16], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $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))))) < 5.0000000000000004e-16

   1. Initial program 73.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Simplified 100.0%

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

   if 5.0000000000000004e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.4%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

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

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

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

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

      13. neg-lowering-neg.f64 N/A

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

      14. distribute-rgt1-in N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto wj - \color{blue}{\mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{-e^{wj} \cdot \left(wj + 1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{e^{wj} \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{-1 - wj}, x \cdot \left(\frac{wj}{x} - e^{-wj}\right), wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 5e-16)
     (fma
      wj
      (fma
       wj
       (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
       (* x -2.0))
      x)
     (fma (/ 1.0 (- -1.0 wj)) (* x (- (/ wj x) (exp (- wj)))) wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 5e-16) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = fma((1.0 / (-1.0 - wj)), (x * ((wj / x) - exp(-wj))), wj);
	}
	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))) <= 5e-16)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = fma(Float64(1.0 / Float64(-1.0 - wj)), Float64(x * Float64(Float64(wj / x) - exp(Float64(-wj)))), wj);
	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], 5e-16], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(wj / x), $MachinePrecision] - N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + wj), $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))))) < 5.0000000000000004e-16

   1. Initial program 73.1%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Simplified 100.0%

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

   if 5.0000000000000004e-16 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 95.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-frac N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt1-in N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. +-lowering-+.f64 N/A

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

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

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

      11. --lowering--.f64 N/A

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

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

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

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

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

      14. exp-lowering-exp.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{wj + 1}, \color{blue}{x \cdot \left(\frac{wj}{x} - \frac{1}{e^{wj}}\right)}, wj\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

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

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-frac-neg2 N/A

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

      15. mul-1-neg N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. --lowering--.f64 N/A

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

      19. /-lowering-/.f64 N/A

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

      20. rec-exp N/A

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

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

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

      22. neg-lowering-neg.f64 99.8

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

   7. Simplified 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{wj + 1}, \color{blue}{x \cdot \left(\frac{wj}{x} - e^{-wj}\right)}, wj\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{-1 - wj}, x \cdot \left(\frac{wj}{x} - e^{-wj}\right), wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0029:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(wj, wj \cdot wj, 1\right)}, \frac{wj}{\frac{1}{\mathsf{fma}\left(wj, wj, 1\right) - wj}}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0029)
   (fma
    wj
    (fma
     wj
     (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
     (* x -2.0))
    x)
   (fma
    (/ -1.0 (fma wj (* wj wj) 1.0))
    (/ wj (/ 1.0 (- (fma wj wj 1.0) wj)))
    wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0029) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = fma((-1.0 / fma(wj, (wj * wj), 1.0)), (wj / (1.0 / (fma(wj, wj, 1.0) - wj))), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0029)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = fma(Float64(-1.0 / fma(wj, Float64(wj * wj), 1.0)), Float64(wj / Float64(1.0 / Float64(fma(wj, wj, 1.0) - wj))), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0029], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(-1.0 / N[(wj * N[(wj * wj), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(wj / N[(1.0 / N[(N[(wj * wj + 1.0), $MachinePrecision] - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.0029

   1. Initial program 79.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Simplified 98.8%

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

   if 0.0029 < wj

   1. Initial program 70.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 85.4

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

   5. Simplified 85.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. +-lowering-+.f64 85.4

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

   7. Applied egg-rr 85.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. un-div-inv N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. flip3-+ N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

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

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

   9. Applied egg-rr 85.7%

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

Alternative 4: 97.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.011)
   (fma
    wj
    (fma
     wj
     (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
     (* x -2.0))
    x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.011) {
		tmp = fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.011)
		tmp = fma(wj, fma(wj, Float64(fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, Float64(x * 2.0)), wj)), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.011], N[(wj * N[(wj * N[(N[(x * 2.5 + 1.0), $MachinePrecision] - N[(wj * N[(x * 0.6666666666666666 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.010999999999999999

   1. Initial program 79.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Simplified 98.8%

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

   if 0.010999999999999999 < wj

   1. Initial program 70.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 85.4

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

   5. Simplified 85.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.011:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.0021:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.0021)
   (fma
    x
    (* wj (fma wj (+ (/ (- 1.0 wj) x) (fma wj -2.6666666666666665 2.5)) -2.0))
    x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.0021) {
		tmp = fma(x, (wj * fma(wj, (((1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.0021)
		tmp = fma(x, Float64(wj * fma(wj, Float64(Float64(Float64(1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.0021], N[(x * N[(wj * N[(wj * N[(N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] + N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 0.00209999999999999987

   1. Initial program 79.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(wj \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right) + \frac{{wj}^{2} \cdot \left(1 - wj\right)}{x}\right)\right)} \]

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, wj \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)} \]

   if 0.00209999999999999987 < wj

   1. Initial program 70.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 85.4

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

   5. Simplified 85.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.0021:\\ \;\;\;\;\mathsf{fma}\left(x, wj \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00071:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00071)
   (fma wj (fma x -2.0 (fma (* wj x) 2.5 wj)) x)
   (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00071) {
		tmp = fma(wj, fma(x, -2.0, fma((wj * x), 2.5, wj)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00071)
		tmp = fma(wj, fma(x, -2.0, fma(Float64(wj * x), 2.5, wj)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00071], N[(wj * N[(x * -2.0 + N[(N[(wj * x), $MachinePrecision] * 2.5 + wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 7.10000000000000019e-4

   1. Initial program 79.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

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

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

   5. Simplified 98.5%

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

   if 7.10000000000000019e-4 < wj

   1. Initial program 70.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 85.4

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

   5. Simplified 85.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00071:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(x, -2, \mathsf{fma}\left(wj \cdot x, 2.5, wj\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00052:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00052) (fma wj (- wj (* wj wj)) x) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00052) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00052)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00052], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.19999999999999954e-4

   1. Initial program 79.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in wj around 0

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. unpow2 N/A

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

      6. unsub-neg N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 97.9

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

   8. Simplified 97.9%

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

   if 5.19999999999999954e-4 < wj

   1. Initial program 70.4%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 85.4

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

   5. Simplified 85.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00052:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.6% accurate, 22.1× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 (fma wj (- wj (* wj wj)) x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Simplified 96.3%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. unpow2 N/A

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

   6. unsub-neg N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 95.4

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

8. Simplified 95.4%

   \[\leadsto \mathsf{fma}\left(wj, \color{blue}{wj - wj \cdot wj}, x\right) \]
9. Add Preprocessing

Alternative 9: 95.3% accurate, 47.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, wj, x\right) \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (fma wj wj x))

C

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

Julia

function code(wj, x)
	return fma(wj, wj, x)
end

Wolfram

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

Derivation

1. Initial program 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Simplified 96.3%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. unpow2 N/A

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

   6. unsub-neg N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 95.4

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

8. Simplified 95.4%

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

9. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + {wj}^{2}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{{wj}^{2} + x} \]

    2. unpow2 N/A

       \[\leadsto \color{blue}{wj \cdot wj} + x \]

    3. accelerator-lowering-fma.f64 95.3

       \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x\right)} \]

11. Simplified 95.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x\right)} \]
12. Add Preprocessing

Alternative 10: 83.9% accurate, 331.0× 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 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around 0

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

5. Simplified 84.8%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 11: 4.5% accurate, 331.0× 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 79.1%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
2. Add Preprocessing

3. Taylor expanded in wj around inf

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

5. Simplified 4.6%

   \[\leadsto \color{blue}{wj} \]
6. Add Preprocessing

Developer Target 1: 78.2% accurate, 1.4× 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 2024199 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((ew (exp wj))) (- wj (- (/ wj (+ wj 1)) (/ x (+ ew (* wj ew)))))))

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