Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.7% → 96.0%

Time: 13.4s

Alternatives: 8

Speedup: 313.0×

• could not determine a ground truth

  1. wj = -6.819200294775787e+224
  2. x = 6.885745082131581e+119

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: 77.7% 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: 96.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (+
  x
  (*
   wj
   (+
    (* x -2.0)
    (*
     wj
     (+
      (+
       1.0
       (* wj (- (- -1.0 (* x 0.6666666666666666)) (+ (* x -3.0) (* x 5.0)))))
      (* x 2.5)))))))

C

double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (-2.0d0)) + (wj * ((1.0d0 + (wj * (((-1.0d0) - (x * 0.6666666666666666d0)) - ((x * (-3.0d0)) + (x * 5.0d0))))) + (x * 2.5d0)))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))))

Julia

function code(wj, x)
	return Float64(x + Float64(wj * Float64(Float64(x * -2.0) + Float64(wj * Float64(Float64(1.0 + Float64(wj * Float64(Float64(-1.0 - Float64(x * 0.6666666666666666)) - Float64(Float64(x * -3.0) + Float64(x * 5.0))))) + Float64(x * 2.5))))))
end

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) + (wj * ((1.0 + (wj * ((-1.0 - (x * 0.6666666666666666)) - ((x * -3.0) + (x * 5.0))))) + (x * 2.5)))));
end

Wolfram

code[wj_, x_] := N[(x + N[(wj * N[(N[(x * -2.0), $MachinePrecision] + N[(wj * N[(N[(1.0 + N[(wj * N[(N[(-1.0 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(x * -3.0), $MachinePrecision] + N[(x * 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Final simplification 98.1%

   \[\leadsto x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 + wj \cdot \left(\left(-1 - x \cdot 0.6666666666666666\right) - \left(x \cdot -3 + x \cdot 5\right)\right)\right) + x \cdot 2.5\right)\right) \]
7. Add Preprocessing

Alternative 2: 95.7% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    code = x + (wj * ((x * (-2.0d0)) + (wj * (1.0d0 + (x * 2.5d0)))))
end function

Java

public static double code(double wj, double x) {
	return x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
}

Python

def code(wj, x):
	return x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))))

Julia

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

MATLAB

function tmp = code(wj, x)
	tmp = x + (wj * ((x * -2.0) + (wj * (1.0 + (x * 2.5)))));
end

Wolfram

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

Derivation

1. Initial program 77.4%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(x \cdot -2\right), \left(\color{blue}{wj} \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \left(\color{blue}{wj} \cdot \left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \color{blue}{\left(1 - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \left(1 + \color{blue}{\left(\mathsf{neg}\left(\left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(\left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right)\right)\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(x \cdot \left(-4 + \frac{3}{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(-4 + \frac{3}{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(-4 + \frac{3}{2}\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\frac{-5}{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. metadata-eval 97.5%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \frac{5}{2}\right)\right)\right)\right)\right)\right) \]

5. Simplified 97.5%

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

Alternative 3: 95.3% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left({wj}^{2} \cdot \left(1 - wj\right)\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(\color{blue}{1} - wj\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(1 + -1 \cdot \color{blue}{wj}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(wj \cdot \left(1 + -1 \cdot wj\right)\right)}\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(wj \cdot \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(wj \cdot \left(1 - \color{blue}{wj}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 + -1 \cdot \color{blue}{wj}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \color{blue}{\left(1 + -1 \cdot wj\right)}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \left(1 - \color{blue}{wj}\right)\right)\right)\right) \]

   13. --lowering--.f64 96.8%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right) \]

8. Simplified 96.8%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \color{blue}{\left(1 - wj\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(1 - wj\right) \cdot \color{blue}{\left(wj \cdot wj\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\left(1 - wj\right), \color{blue}{\left(wj \cdot wj\right)}\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, wj\right), \left(\color{blue}{wj} \cdot wj\right)\right)\right) \]

   5. *-lowering-*.f64 96.8%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, wj\right), \mathsf{*.f64}\left(wj, \color{blue}{wj}\right)\right)\right) \]

10. Applied egg-rr 96.8%

    \[\leadsto x + \color{blue}{\left(1 - wj\right) \cdot \left(wj \cdot wj\right)} \]
11. Add Preprocessing

Alternative 4: 95.3% accurate, 34.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left({wj}^{2} \cdot \left(1 - wj\right)\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(\color{blue}{1} - wj\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(1 + -1 \cdot \color{blue}{wj}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(wj \cdot \left(1 + -1 \cdot wj\right)\right)}\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(wj \cdot \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(wj \cdot \left(1 - \color{blue}{wj}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 + -1 \cdot \color{blue}{wj}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \color{blue}{\left(1 + -1 \cdot wj\right)}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \left(1 - \color{blue}{wj}\right)\right)\right)\right) \]

   13. --lowering--.f64 96.8%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right) \]

8. Simplified 96.8%

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

Alternative 5: 82.8% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 5.8 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;wj \cdot wj\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x) :precision binary64 (if (<= wj 5.8e-60) x (* wj wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 5.8e-60) {
		tmp = x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: tmp
    if (wj <= 5.8d-60) then
        tmp = x
    else
        tmp = wj * wj
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double tmp;
	if (wj <= 5.8e-60) {
		tmp = x;
	} else {
		tmp = wj * wj;
	}
	return tmp;
}

Python

def code(wj, x):
	tmp = 0
	if wj <= 5.8e-60:
		tmp = x
	else:
		tmp = wj * wj
	return tmp

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 5.8e-60)
		tmp = x;
	else
		tmp = Float64(wj * wj);
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	tmp = 0.0;
	if (wj <= 5.8e-60)
		tmp = x;
	else
		tmp = wj * wj;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 5.8e-60], x, N[(wj * wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 5.7999999999999999e-60

   1. Initial program 82.4%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}} \]
   4. Add Preprocessing

   5. Taylor expanded in wj around 0

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 89.2%

      \[\leadsto \color{blue}{x} \]

   if 5.7999999999999999e-60 < wj

   1. Initial program 37.1%

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

      3. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

      4. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      9. *-inverses N/A

         \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

      13. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

      15. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

      16. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. Simplified 44.1%

      \[\leadsto \color{blue}{wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(wj, \color{blue}{\left(\frac{wj}{1 + wj}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(wj, \color{blue}{\left(1 + wj\right)}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(wj, \left(wj + \color{blue}{1}\right)\right)\right) \]

      6. +-lowering-+.f64 13.0%

         \[\leadsto \mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(wj, \mathsf{+.f64}\left(wj, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 13.0%

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

   8. Taylor expanded in wj around 0

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

      1. unpow2 N/A

         \[\leadsto wj \cdot \color{blue}{wj} \]

      2. *-lowering-*.f64 58.1%

         \[\leadsto \mathsf{*.f64}\left(wj, \color{blue}{wj}\right) \]

   10. Simplified 58.1%

       \[\leadsto \color{blue}{wj \cdot wj} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.0% accurate, 62.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in wj around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(wj \cdot \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) - 2 \cdot x\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right) + -2 \cdot x\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(-2 \cdot x + \color{blue}{wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{+.f64}\left(\left(-2 \cdot x\right), \color{blue}{\left(wj \cdot \left(\left(1 + -1 \cdot \left(wj \cdot \left(1 + \left(-3 \cdot x + \left(-2 \cdot \left(-4 \cdot x + \frac{3}{2} \cdot x\right) + \frac{2}{3} \cdot x\right)\right)\right)\right)\right) - \left(-4 \cdot x + \frac{3}{2} \cdot x\right)\right)\right)}\right)\right)\right) \]

5. Simplified 98.1%

   \[\leadsto \color{blue}{x + wj \cdot \left(x \cdot -2 + wj \cdot \left(\left(1 - wj \cdot \left(\left(x \cdot -3 + x \cdot 5\right) + \left(x \cdot 0.6666666666666666 + 1\right)\right)\right) + x \cdot 2.5\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left({wj}^{2} \cdot \left(1 - wj\right)\right)}\right) \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(\color{blue}{1} - wj\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(\left(wj \cdot wj\right) \cdot \left(1 + -1 \cdot \color{blue}{wj}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \color{blue}{\left(wj \cdot \left(1 + -1 \cdot wj\right)\right)}\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(wj \cdot \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(wj \cdot \left(wj \cdot \left(1 - \color{blue}{wj}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \color{blue}{\left(wj \cdot \left(1 - wj\right)\right)}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \left(wj \cdot \left(1 + -1 \cdot \color{blue}{wj}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \color{blue}{\left(1 + -1 \cdot wj\right)}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \left(1 + \left(\mathsf{neg}\left(wj\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \left(1 - \color{blue}{wj}\right)\right)\right)\right) \]

   13. --lowering--.f64 96.8%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(wj, \mathsf{*.f64}\left(wj, \mathsf{\_.f64}\left(1, \color{blue}{wj}\right)\right)\right)\right) \]

8. Simplified 96.8%

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

9. Taylor expanded in wj around 0

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

    1. +-commutative N/A

       \[\leadsto {wj}^{2} + \color{blue}{x} \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left({wj}^{2}\right), \color{blue}{x}\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(wj \cdot wj\right), x\right) \]

    4. *-lowering-*.f64 96.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(wj, wj\right), x\right) \]

11. Simplified 96.4%

    \[\leadsto \color{blue}{wj \cdot wj + x} \]

12. Final simplification 96.4%

    \[\leadsto x + wj \cdot wj \]
13. Add Preprocessing

Alternative 7: 83.9% accurate, 313.0× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(wj, x):
	return x

Julia

function code(wj, x)
	return x
end

MATLAB

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

Wolfram

code[wj_, x_] := x

Derivation

1. Initial program 77.4%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 78.6%

   \[\leadsto \color{blue}{wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}} \]
4. Add Preprocessing

5. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 82.7%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Alternative 8: 4.4% accurate, 313.0× 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 wj
end

MATLAB

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

Wolfram

code[wj_, x_] := wj

Derivation

1. Initial program 77.4%

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

   1. sub-neg N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \color{blue}{\left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)\right)}\right) \]

   3. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj} - x}{\left(wj + 1\right) \cdot e^{wj}}\right)\right)\right) \]

   4. associate-/l/ N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\mathsf{neg}\left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{wj + 1}\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \left(\frac{\frac{wj \cdot e^{wj} - x}{e^{wj}}}{\color{blue}{\mathsf{neg}\left(\left(wj + 1\right)\right)}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj} - x}{e^{wj}}\right), \color{blue}{\left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(\frac{wj \cdot e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot \frac{e^{wj}}{e^{wj}} - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   9. *-inverses N/A

      \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj \cdot 1 - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\left(wj - \frac{x}{e^{wj}}\right), \left(\mathsf{neg}\left(\left(\color{blue}{wj} + 1\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \left(\frac{x}{e^{wj}}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\left(wj + 1\right)}\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \left(e^{wj}\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + \color{blue}{1}\right)\right)\right)\right)\right) \]

   13. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(wj + 1\right)\right)\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(1 + wj\right)\right)\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\mathsf{neg}\left(\left(\frac{e^{wj}}{e^{wj}} + wj\right)\right)\right)\right)\right) \]

   16. distribute-neg-in N/A

       \[\leadsto \mathsf{+.f64}\left(wj, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(wj, \mathsf{/.f64}\left(x, \mathsf{exp.f64}\left(wj\right)\right)\right), \left(\left(\mathsf{neg}\left(\frac{e^{wj}}{e^{wj}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(wj\right)\right)}\right)\right)\right) \]

3. Simplified 78.6%

   \[\leadsto \color{blue}{wj + \frac{wj - \frac{x}{e^{wj}}}{-1 - wj}} \]
4. Add Preprocessing

5. Taylor expanded in wj around inf

   \[\leadsto \color{blue}{wj} \]
6. Step-by-step derivation

7. Simplified 4.4%

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

Developer Target 1: 78.7% accurate, 1.5× 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 2024164 
(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))))))