Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.3% → 96.3%

Time: 8.4s

Alternatives: 10

Speedup: 55.2×

• could not determine a ground truth

  1. wj = 4.37164697170927e+116
  2. x = 8.603583414383355e-63

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: 78.3% 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: 96.3% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 97.6%

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

Alternative 2: 96.3% 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 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.3%

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

Alternative 3: 95.6% 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 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 96.8%

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

Alternative 4: 85.0% 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 80.9%

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

3. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x + -2 \cdot \left(wj \cdot x\right)} \]
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 85.7

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

5. Applied rewrites 85.7%

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

Alternative 5: 85.0% 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 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.9%

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

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 85.7

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

8. Applied rewrites 85.7%

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

Alternative 6: 84.5% 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 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 96.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 97.6%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 85.3%

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

Alternative 7: 13.6% accurate, 55.2× speedup

Math

\[\begin{array}{l} \\ wj \cdot wj \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{-5}{2}}\right), x, 1\right), wj, -2 \cdot x\right), wj, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{5}{2}}, x, 1\right), wj, -2 \cdot x\right), wj, x\right) \]

   16. lower-*.f64 96.5

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

5. Applied rewrites 96.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 13.8%

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

Alternative 8: 4.3% 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 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 3.8

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

5. Applied rewrites 3.8%

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

Alternative 9: 3.9% accurate, 110.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return -wj

Julia

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

MATLAB

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

Wolfram

code[wj_, x_] := (-wj)

Derivation

1. Initial program 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 3.8

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

5. Applied rewrites 3.8%

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

7. Applied rewrites 3.3%

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

8. Taylor expanded in wj around inf

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

10. Applied rewrites 3.7%

    \[\leadsto -wj \]
11. Add Preprocessing

Alternative 10: 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 80.9%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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 3.8

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

5. Applied rewrites 3.8%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 3.4%

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

Developer Target 1: 79.3% 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 2024288 
(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))))))