Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 98.7%

Time: 7.9s

Alternatives: 9

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 8.437960932887736e+41
  2. x = -1.0204541362874851e+204

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: 77.8% 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: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), x, \left(1 - wj\right) \cdot wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, \left(\frac{wj}{x} - e^{-wj}\right) \cdot x, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-15)
     (fma
      (fma (fma (fma -2.6666666666666665 wj 2.5) wj -2.0) x (* (- 1.0 wj) wj))
      wj
      x)
     (fma (/ -1.0 (+ 1.0 wj)) (* (- (/ wj x) (exp (- wj))) x) wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e-15) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), x, ((1.0 - wj) * wj)), wj, x);
	} else {
		tmp = fma((-1.0 / (1.0 + wj)), (((wj / x) - exp(-wj)) * x), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-15)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), x, Float64(Float64(1.0 - wj) * wj)), wj, x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 + wj)), Float64(Float64(Float64(wj / x) - exp(Float64(-wj))) * x), 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[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision] * x + N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(wj / x), $MachinePrecision] - N[Exp[(-wj)], $MachinePrecision]), $MachinePrecision] * x), $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))))) < 4.99999999999999999e-15

   1. Initial program 69.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right) + x \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right), wj, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.0%

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

   if 4.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj - \frac{t\_0 - x}{e^{wj} + t\_0} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), x, \left(1 - wj\right) \cdot wj\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{1 + wj}, wj - e^{-wj} \cdot x, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (- wj (/ (- t_0 x) (+ (exp wj) t_0))) 5e-15)
     (fma
      (fma (fma (fma -2.6666666666666665 wj 2.5) wj -2.0) x (* (- 1.0 wj) wj))
      wj
      x)
     (fma (/ -1.0 (+ 1.0 wj)) (- wj (* (exp (- wj)) x)) wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj - ((t_0 - x) / (exp(wj) + t_0))) <= 5e-15) {
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), x, ((1.0 - wj) * wj)), wj, x);
	} else {
		tmp = fma((-1.0 / (1.0 + wj)), (wj - (exp(-wj) * x)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0))) <= 5e-15)
		tmp = fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), x, Float64(Float64(1.0 - wj) * wj)), wj, x);
	else
		tmp = fma(Float64(-1.0 / Float64(1.0 + wj)), Float64(wj - Float64(exp(Float64(-wj)) * x)), 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[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-15], N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * wj + -2.0), $MachinePrecision] * x + N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(-1.0 / N[(1.0 + wj), $MachinePrecision]), $MachinePrecision] * N[(wj - N[(N[Exp[(-wj)], $MachinePrecision] * x), $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))))) < 4.99999999999999999e-15

   1. Initial program 69.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 99.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right) + x \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right), wj, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.0%

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

   if 4.99999999999999999e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 99.5

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.5%

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

Alternative 3: 96.1% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), x, \left(1 - wj\right) \cdot wj\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma (fma (fma -2.6666666666666665 wj 2.5) wj -2.0) x (* (- 1.0 wj) wj))
  wj
  x))

C

double code(double wj, double x) {
	return fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), x, ((1.0 - wj) * wj)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(fma(fma(-2.6666666666666665, wj, 2.5), wj, -2.0), x, Float64(Float64(1.0 - wj) * wj)), wj, x)
end

Wolfram

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

Derivation

1. Initial program 76.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(wj \cdot \left(1 - wj\right) + x \cdot \left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) - 2\right), wj, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), wj, -2\right), x, \left(1 - wj\right) \cdot wj\right), wj, x\right) \]
9. Add Preprocessing

Alternative 4: 95.3% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(Float64(Float64(1.0 - wj) * wj), wj, x)
end

Wolfram

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

Derivation

1. Initial program 76.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.9%

   \[\leadsto \mathsf{fma}\left(\left(1 - wj\right) \cdot wj, wj, x\right) \]
9. Add Preprocessing

Alternative 5: 84.1% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 85.5

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

5. Applied rewrites 85.5%

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

Alternative 6: 84.1% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return Float64(fma(-2.0, wj, 1.0) * x)
end

Wolfram

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

Derivation

1. Initial program 76.2%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.3%

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

6. Taylor expanded in wj around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. +-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 85.5

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

8. Applied rewrites 85.5%

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

Alternative 7: 72.5% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return wj - -x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in wj around 0

   \[\leadsto wj - \color{blue}{-1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 69.2

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

5. Applied rewrites 69.2%

   \[\leadsto wj - \color{blue}{\left(-x\right)} \]
6. Add Preprocessing

Alternative 8: 4.4% accurate, 82.8× speedup

Math

\[\begin{array}{l} \\ -1 + wj \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (+ -1.0 wj))

C

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

Fortran

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

Java

public static double code(double wj, double x) {
	return -1.0 + wj;
}

Python

def code(wj, x):
	return -1.0 + wj

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 4.0

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

5. Applied rewrites 4.0%

   \[\leadsto \color{blue}{-1 + wj} \]
6. Add Preprocessing

Alternative 9: 3.4% accurate, 331.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 -1.0)

C

double code(double wj, double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double wj, double x) {
	return -1.0;
}

Python

def code(wj, x):
	return -1.0

Julia

function code(wj, x)
	return -1.0
end

MATLAB

function tmp = code(wj, x)
	tmp = -1.0;
end

Wolfram

code[wj_, x_] := -1.0

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in wj around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. lft-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 4.0

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in wj around 0

   \[\leadsto -1 \]
7. Step-by-step derivation

8. Applied rewrites 3.5%

   \[\leadsto -1 \]
9. Add Preprocessing

Developer Target 1: 78.8% 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 2024308 
(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))))))