Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.6% → 98.0%

Time: 7.7s

Alternatives: 7

Speedup: 55.2×

• could not determine a ground truth

  1. wj = 3.155414088588453e+138
  2. x = -1.5764032473639155e-58

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: 78.6% 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: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 80.4%

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

7. Applied rewrites 79.4%

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

8. Taylor expanded in x around -inf

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

10. Applied rewrites 90.8%

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

11. Taylor expanded in wj around 0

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

13. Applied rewrites 99.3%

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

14. Final simplification 99.3%

    \[\leadsto \left(\left(\mathsf{fma}\left(\frac{wj}{x} - {x}^{-1}, wj, {x}^{-1}\right) \cdot wj - {x}^{-1}\right) \cdot \left(wj \cdot wj\right) - \frac{e^{-wj}}{1 + wj}\right) \cdot \left(-x\right) \]
15. Add Preprocessing

Alternative 2: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(-0.5 \cdot wj - 1\right) \cdot wj - 1, wj \cdot wj, -x\right)}{x \cdot \left(\left(-1 - wj\right) \cdot e^{wj}\right)} \cdot x \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (*
  (/
   (fma (- (* (- (* -0.5 wj) 1.0) wj) 1.0) (* wj wj) (- x))
   (* x (* (- -1.0 wj) (exp wj))))
  x))

C

double code(double wj, double x) {
	return (fma(((((-0.5 * wj) - 1.0) * wj) - 1.0), (wj * wj), -x) / (x * ((-1.0 - wj) * exp(wj)))) * x;
}

Julia

function code(wj, x)
	return Float64(Float64(fma(Float64(Float64(Float64(Float64(-0.5 * wj) - 1.0) * wj) - 1.0), Float64(wj * wj), Float64(-x)) / Float64(x * Float64(Float64(-1.0 - wj) * exp(wj)))) * x)
end

Wolfram

code[wj_, x_] := N[(N[(N[(N[(N[(N[(N[(-0.5 * wj), $MachinePrecision] - 1.0), $MachinePrecision] * wj), $MachinePrecision] - 1.0), $MachinePrecision] * N[(wj * wj), $MachinePrecision] + (-x)), $MachinePrecision] / N[(x * N[(N[(-1.0 - wj), $MachinePrecision] * N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 80.4%

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

7. Applied rewrites 79.4%

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

8. Taylor expanded in wj around 0

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

10. Applied rewrites 98.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-0.5 \cdot wj - 1\right) \cdot wj - 1, wj \cdot wj, -x\right)}{x \cdot \left(\left(-1 + \left(-wj\right)\right) \cdot e^{wj}\right)} \cdot x \]

11. Final simplification 98.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-0.5 \cdot wj - 1\right) \cdot wj - 1, wj \cdot wj, -x\right)}{x \cdot \left(\left(-1 - wj\right) \cdot e^{wj}\right)} \cdot x \]
12. Add Preprocessing

Alternative 3: 96.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \mathsf{fma}\left(\mathsf{fma}\left(x, 2.3333333333333335, \mathsf{fma}\left(-5, x, -2\right)\right), wj, wj\right)\right), wj, wj\right)\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma
   -2.0
   x
   (fma
    (fma 2.5 x (fma (fma x 2.3333333333333335 (fma -5.0 x -2.0)) wj wj))
    wj
    wj))
  wj
  x))

C

double code(double wj, double x) {
	return fma(fma(-2.0, x, fma(fma(2.5, x, fma(fma(x, 2.3333333333333335, fma(-5.0, x, -2.0)), wj, wj)), wj, wj)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(-2.0, x, fma(fma(2.5, x, fma(fma(x, 2.3333333333333335, fma(-5.0, x, -2.0)), wj, wj)), wj, wj)), wj, x)
end

Wolfram

code[wj_, x_] := N[(N[(-2.0 * x + N[(N[(2.5 * x + N[(N[(x * 2.3333333333333335 + N[(-5.0 * x + -2.0), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * wj + wj), $MachinePrecision]), $MachinePrecision] * 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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 80.4%

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

7. Applied rewrites 79.4%

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

8. Taylor expanded in wj around 0

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

10. Applied rewrites 98.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(-0.5 \cdot wj - 1\right) \cdot wj - 1, wj \cdot wj, -x\right)}{x \cdot \left(\left(-1 + \left(-wj\right)\right) \cdot e^{wj}\right)} \cdot x \]

11. Taylor expanded in wj around 0

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

13. Applied rewrites 98.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-2, x, \mathsf{fma}\left(\mathsf{fma}\left(2.5, x, \mathsf{fma}\left(\mathsf{fma}\left(x, 2.3333333333333335, \mathsf{fma}\left(-5, x, -2\right)\right), wj, wj\right)\right), wj, wj\right)\right), \color{blue}{wj}, x\right) \]
14. Add Preprocessing

Alternative 4: 95.9% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[wj_, x_] := N[(N[(N[(1.0 - wj), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * 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. Applied rewrites 98.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

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

Alternative 5: 94.9% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[wj_, x_] := N[(N[(N[(2.5 * x + 1.0), $MachinePrecision] * wj), $MachinePrecision] * 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. Applied rewrites 98.1%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 97.3%

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

9. Taylor expanded in wj around inf

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

11. Applied rewrites 72.7%

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

12. Taylor expanded in wj around 0

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

14. Applied rewrites 96.7%

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

Alternative 6: 84.8% 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 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

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

   1. associate-*r* N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt1-in N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 87.3

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

5. Applied rewrites 87.3%

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

Alternative 7: 84.2% accurate, 55.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(1.0 * 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 x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 86.9%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Developer Target 1: 79.6% 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 2024326 
(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))))))