Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 98.1%

Time: 8.9s

Alternatives: 13

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: 98.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. times-frac N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f64 N/A

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

8. Applied rewrites 90.7%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 98.3%

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

Alternative 2: 97.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. times-frac N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f64 N/A

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

8. Applied rewrites 90.7%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 98.0%

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

Alternative 3: 97.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-rgt-identity N/A

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. times-frac N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-fma.f64 N/A

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

8. Applied rewrites 90.7%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 97.3%

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

Alternative 4: 96.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \left(\mathsf{fma}\left(7.111111111111111, wj \cdot wj, -6.25\right) \cdot x\right) \cdot \left(1 + wj\right)\right)}{\mathsf{fma}\left(-5.166666666666667, wj, -2.5\right)}, wj, -2 \cdot x\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma
   (/
    (fma
     (- 1.0 (* wj wj))
     (fma -2.6666666666666665 wj -2.5)
     (* (* (fma 7.111111111111111 (* wj wj) -6.25) x) (+ 1.0 wj)))
    (fma -5.166666666666667 wj -2.5))
   wj
   (* -2.0 x))
  wj
  x))

C

double code(double wj, double x) {
	return fma(fma((fma((1.0 - (wj * wj)), fma(-2.6666666666666665, wj, -2.5), ((fma(7.111111111111111, (wj * wj), -6.25) * x) * (1.0 + wj))) / fma(-5.166666666666667, wj, -2.5)), wj, (-2.0 * x)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(Float64(fma(Float64(1.0 - Float64(wj * wj)), fma(-2.6666666666666665, wj, -2.5), Float64(Float64(fma(7.111111111111111, Float64(wj * wj), -6.25) * x) * Float64(1.0 + wj))) / fma(-5.166666666666667, wj, -2.5)), wj, Float64(-2.0 * x)), wj, x)
end

Wolfram

code[wj_, x_] := N[(N[(N[(N[(N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision] * N[(-2.6666666666666665 * wj + -2.5), $MachinePrecision] + N[(N[(N[(7.111111111111111 * N[(wj * wj), $MachinePrecision] + -6.25), $MachinePrecision] * x), $MachinePrecision] * N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-5.166666666666667 * wj + -2.5), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $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 wj around 0

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

8. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \left(1 + wj\right) \cdot \left(\mathsf{fma}\left(7.111111111111111, wj \cdot wj, -6.25\right) \cdot x\right)\right)}{\left(1 + wj\right) \cdot \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right)}, wj, -2 \cdot x\right), wj, x\right) \]

11. Taylor expanded in wj around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(\frac{-8}{3}, wj, \frac{-5}{2}\right), \left(1 + wj\right) \cdot \left(\mathsf{fma}\left(\frac{64}{9}, wj \cdot wj, \frac{-25}{4}\right) \cdot x\right)\right)}{\frac{-31}{6} \cdot wj - \frac{5}{2}}, wj, -2 \cdot x\right), wj, x\right) \]
12. Step-by-step derivation

13. Applied rewrites 96.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \left(1 + wj\right) \cdot \left(\mathsf{fma}\left(7.111111111111111, wj \cdot wj, -6.25\right) \cdot x\right)\right)}{\mathsf{fma}\left(-5.166666666666667, wj, -2.5\right)}, wj, -2 \cdot x\right), wj, x\right) \]

14. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \left(\mathsf{fma}\left(7.111111111111111, wj \cdot wj, -6.25\right) \cdot x\right) \cdot \left(1 + wj\right)\right)}{\mathsf{fma}\left(-5.166666666666667, wj, -2.5\right)}, wj, -2 \cdot x\right), wj, x\right) \]
15. Add Preprocessing

Alternative 5: 96.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \mathsf{fma}\left(x, wj, x\right) \cdot -6.25\right)}{\mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right) \cdot \left(1 + wj\right)}, wj, -2 \cdot x\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma
   (/
    (fma
     (- 1.0 (* wj wj))
     (fma -2.6666666666666665 wj -2.5)
     (* (fma x wj x) -6.25))
    (* (fma -2.6666666666666665 wj -2.5) (+ 1.0 wj)))
   wj
   (* -2.0 x))
  wj
  x))

C

double code(double wj, double x) {
	return fma(fma((fma((1.0 - (wj * wj)), fma(-2.6666666666666665, wj, -2.5), (fma(x, wj, x) * -6.25)) / (fma(-2.6666666666666665, wj, -2.5) * (1.0 + wj))), wj, (-2.0 * x)), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(Float64(fma(Float64(1.0 - Float64(wj * wj)), fma(-2.6666666666666665, wj, -2.5), Float64(fma(x, wj, x) * -6.25)) / Float64(fma(-2.6666666666666665, wj, -2.5) * Float64(1.0 + wj))), wj, Float64(-2.0 * x)), wj, x)
end

Wolfram

code[wj_, x_] := N[(N[(N[(N[(N[(1.0 - N[(wj * wj), $MachinePrecision]), $MachinePrecision] * N[(-2.6666666666666665 * wj + -2.5), $MachinePrecision] + N[(N[(x * wj + x), $MachinePrecision] * -6.25), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.6666666666666665 * wj + -2.5), $MachinePrecision] * N[(1.0 + wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $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 wj around 0

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

8. Applied rewrites 96.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6666666666666665, wj, 2.5\right), x, 1 - wj\right), wj, -2 \cdot x\right), wj, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \left(1 + wj\right) \cdot \left(\mathsf{fma}\left(7.111111111111111, wj \cdot wj, -6.25\right) \cdot x\right)\right)}{\left(1 + wj\right) \cdot \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right)}, wj, -2 \cdot x\right), wj, x\right) \]

11. Taylor expanded in wj around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(\frac{-8}{3}, wj, \frac{-5}{2}\right), \frac{-25}{4} \cdot x + \frac{-25}{4} \cdot \left(wj \cdot x\right)\right)}{\left(1 + wj\right) \cdot \mathsf{fma}\left(\frac{-8}{3}, wj, \frac{-5}{2}\right)}, wj, -2 \cdot x\right), wj, x\right) \]
12. Step-by-step derivation

13. Applied rewrites 96.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), -6.25 \cdot \mathsf{fma}\left(x, wj, x\right)\right)}{\left(1 + wj\right) \cdot \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right)}, wj, -2 \cdot x\right), wj, x\right) \]

14. Final simplification 96.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - wj \cdot wj, \mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right), \mathsf{fma}\left(x, wj, x\right) \cdot -6.25\right)}{\mathsf{fma}\left(-2.6666666666666665, wj, -2.5\right) \cdot \left(1 + wj\right)}, wj, -2 \cdot x\right), wj, x\right) \]
15. Add Preprocessing

Alternative 6: 96.1% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[wj_, x_] := N[(N[(N[(N[(-2.6666666666666665 * wj + 2.5), $MachinePrecision] * x + N[(1.0 - wj), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $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 wj around 0

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

8. Applied rewrites 96.3%

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

Alternative 7: 95.8% 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 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

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

8. Applied rewrites 96.1%

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

Alternative 8: 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 9: 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 10: 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 11: 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 12: 4.4% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := N[(wj - 1.0), $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. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 4.0

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

5. Applied rewrites 4.0%

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

6. Final simplification 4.0%

   \[\leadsto wj - 1 \]
7. Add Preprocessing

Alternative 13: 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. +-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

   5. rgt-mult-inverse N/A

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

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   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))))))