Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.9% → 98.2%

Time: 12.6s

Alternatives: 9

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 7.651187167493154e+121
  2. x = 170488.4559264945

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: 77.9% 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: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{t\_0 + e^{wj}} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{e^{-wj}}{-1 - wj}\right) - \frac{wj}{\mathsf{fma}\left(wj, x, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ t_0 (exp wj)))) 5e-24)
     (fma
      wj
      (fma
       wj
       (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
       (* x -2.0))
      x)
     (* x (- (- (/ wj x) (/ (exp (- wj)) (- -1.0 wj))) (/ wj (fma wj x x)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (t_0 + exp(wj)))) <= 5e-24) {
		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 = x * (((wj / x) - (exp(-wj) / (-1.0 - wj))) - (wj / fma(wj, x, x)));
	}
	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(t_0 + exp(wj)))) <= 5e-24)
		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(x * Float64(Float64(Float64(wj / x) - Float64(exp(Float64(-wj)) / Float64(-1.0 - wj))) - Float64(wj / fma(wj, x, x))));
	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[(t$95$0 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-24], 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[(x * N[(N[(N[(wj / x), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(wj / N[(wj * x + x), $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))))) < 4.9999999999999998e-24

   1. Initial program 66.0%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Applied rewrites 98.9%

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

   if 4.9999999999999998e-24 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 97.1%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. rec-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. *-commutative N/A

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

      19. *-rgt-identity N/A

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{wj \cdot e^{wj} + e^{wj}} \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\frac{wj}{x} - \frac{e^{-wj}}{-1 - wj}\right) - \frac{wj}{\mathsf{fma}\left(wj, x, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(wj, \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 2.5, 1\right) - \mathsf{fma}\left(wj, \mathsf{fma}\left(x, 0.6666666666666666, x \cdot 2\right), wj\right), x \cdot -2\right), x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  wj
  (fma
   wj
   (- (fma x 2.5 1.0) (fma wj (fma x 0.6666666666666666 (* x 2.0)) wj))
   (* x -2.0))
  x))

C

double code(double wj, double x) {
	return fma(wj, fma(wj, (fma(x, 2.5, 1.0) - fma(wj, fma(x, 0.6666666666666666, (x * 2.0)), wj)), (x * -2.0)), x);
}

Julia

function code(wj, x)
	return 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)
end

Wolfram

code[wj_, x_] := 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]

Derivation

1. Initial program 77.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. Applied rewrites 96.2%

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

Developer Target 1: 78.9% 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 2024219 
(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))))))