Jmat.Real.lambertw, newton loop step

Percentage Accurate: 77.8% → 98.4%

Time: 11.1s

Alternatives: 11

Speedup: 331.0×

• could not determine a ground truth

  1. wj = 1.6607956168381566e+40
  2. x = 5.769358818889093e+58

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ wj - \frac{t\_0 - x}{e^{wj} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))) (- wj (/ (- t_0 x) (+ (exp wj) t_0)))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	return wj - ((t_0 - x) / (exp(wj) + t_0));
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = wj * exp(wj)
    code = wj - ((t_0 - x) / (exp(wj) + t_0))
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	return wj - ((t_0 - x) / (Math.exp(wj) + t_0));
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	return wj - ((t_0 - x) / (math.exp(wj) + t_0))

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	return Float64(wj - Float64(Float64(t_0 - x) / Float64(exp(wj) + t_0)))
end

MATLAB

function tmp = code(wj, x)
	t_0 = wj * exp(wj);
	tmp = wj - ((t_0 - x) / (exp(wj) + t_0));
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, N[(wj - N[(N[(t$95$0 - x), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}, -x, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 4e-26)
     (fma
      wj
      (* x (fma wj (+ (/ (- 1.0 wj) x) (fma wj -2.6666666666666665 2.5)) -2.0))
      x)
     (fma (- (/ wj (fma x wj x)) (/ (exp (- wj)) (+ wj 1.0))) (- x) wj))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 4e-26) {
		tmp = fma(wj, (x * fma(wj, (((1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	} else {
		tmp = fma(((wj / fma(x, wj, x)) - (exp(-wj) / (wj + 1.0))), -x, wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 4e-26)
		tmp = fma(wj, Float64(x * fma(wj, Float64(Float64(Float64(1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	else
		tmp = fma(Float64(Float64(wj / fma(x, wj, x)) - Float64(exp(Float64(-wj)) / Float64(wj + 1.0))), Float64(-x), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-26], N[(wj * N[(x * N[(wj * N[(N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] + N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(wj / N[(x * wj + x), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[(-wj)], $MachinePrecision] / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x) + wj), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.0000000000000002e-26

   1. Initial program 64.4%

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

   3. Taylor expanded in wj around 0

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{fma}\left(wj, \color{blue}{x \cdot \left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \frac{wj \cdot \left(1 - wj\right)}{x}\right) - 2\right)}, x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \color{blue}{\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \frac{wj \cdot \left(1 - wj\right)}{x}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}, x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\color{blue}{\left(\frac{wj \cdot \left(1 - wj\right)}{x} + wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(\color{blue}{wj \cdot \frac{1 - wj}{x}} + wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(wj \cdot \frac{1 - wj}{x} + wj \cdot \color{blue}{\left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)}\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(wj \cdot \frac{1 - wj}{x} + wj \cdot \left(\frac{5}{2} + \color{blue}{\frac{-8}{3}} \cdot wj\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\color{blue}{wj \cdot \left(\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(wj \cdot \left(\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right) + \color{blue}{-2}\right), x\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \color{blue}{\mathsf{fma}\left(wj, \frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right)}, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \color{blue}{\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)}, -2\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \color{blue}{\frac{1 - wj}{x}} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{\color{blue}{1 - wj}}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \color{blue}{\left(\frac{-8}{3} \cdot wj + \frac{5}{2}\right)}, -2\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \left(\color{blue}{wj \cdot \frac{-8}{3}} + \frac{5}{2}\right), -2\right), x\right) \]

      15. accelerator-lowering-fma.f64 98.7

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

   8. Simplified 98.7%

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

   if 4.0000000000000002e-26 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto wj - x \cdot \left(\color{blue}{\left(0 - \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)} + \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right) \]

      4. associate-+l- N/A

         \[\leadsto wj - x \cdot \color{blue}{\left(0 - \left(\frac{1}{e^{wj} + wj \cdot e^{wj}} - \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)} \]

      5. unsub-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \left(\mathsf{neg}\left(\frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}\right)\right)\right)}\right) \]

      6. mul-1-neg N/A

         \[\leadsto wj - x \cdot \left(0 - \left(\frac{1}{e^{wj} + wj \cdot e^{wj}} + \color{blue}{-1 \cdot \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)}}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto wj - x \cdot \left(0 - \color{blue}{\left(-1 \cdot \frac{wj \cdot e^{wj}}{x \cdot \left(e^{wj} + wj \cdot e^{wj}\right)} + \frac{1}{e^{wj} + wj \cdot e^{wj}}\right)}\right) \]

   5. Simplified 99.3%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{wj}{\mathsf{fma}\left(x, wj, x\right)} - \frac{e^{-wj}}{wj + 1}, -x, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t\_0}{e^{wj} + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))) (t_1 (+ wj (/ (- x t_0) (+ (exp wj) t_0)))))
   (if (<= t_1 -1e-303) x (if (<= t_1 0.0) (* wj wj) x))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-303) {
		tmp = x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(wj, x)
    real(8), intent (in) :: wj
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj + ((x - t_0) / (exp(wj) + t_0))
    if (t_1 <= (-1d-303)) then
        tmp = x
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double wj, double x) {
	double t_0 = wj * Math.exp(wj);
	double t_1 = wj + ((x - t_0) / (Math.exp(wj) + t_0));
	double tmp;
	if (t_1 <= -1e-303) {
		tmp = x;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj + ((x - t_0) / (math.exp(wj) + t_0))
	tmp = 0
	if t_1 <= -1e-303:
		tmp = x
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = x
	return tmp

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	t_1 = Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0)))
	tmp = 0.0
	if (t_1 <= -1e-303)
		tmp = x;
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(wj, x)
	t_0 = wj * exp(wj);
	t_1 = wj + ((x - t_0) / (exp(wj) + t_0));
	tmp = 0.0;
	if (t_1 <= -1e-303)
		tmp = x;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-303], x, If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], x]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < -9.99999999999999931e-304 or 0.0 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.7%

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

   3. Taylor expanded in wj around 0

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

   5. Simplified 91.4%

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

   if -9.99999999999999931e-304 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 5.6

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

   5. Simplified 5.6%

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

   6. Taylor expanded in wj around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 58.4

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

   8. Simplified 58.4%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -1 \cdot 10^{-303}:\\ \;\;\;\;x\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{e^{wj} \cdot \left(-1 - wj\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 4e-26)
     (fma
      wj
      (* x (fma wj (+ (/ (- 1.0 wj) x) (fma wj -2.6666666666666665 2.5)) -2.0))
      x)
     (- wj (fma wj (/ 1.0 (+ wj 1.0)) (/ x (* (exp wj) (- -1.0 wj))))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 4e-26) {
		tmp = fma(wj, (x * fma(wj, (((1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	} else {
		tmp = wj - fma(wj, (1.0 / (wj + 1.0)), (x / (exp(wj) * (-1.0 - wj))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = Float64(wj * exp(wj))
	tmp = 0.0
	if (Float64(wj + Float64(Float64(x - t_0) / Float64(exp(wj) + t_0))) <= 4e-26)
		tmp = fma(wj, Float64(x * fma(wj, Float64(Float64(Float64(1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	else
		tmp = Float64(wj - fma(wj, Float64(1.0 / Float64(wj + 1.0)), Float64(x / Float64(exp(wj) * Float64(-1.0 - wj)))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[(wj * N[Exp[wj], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj + N[(N[(x - t$95$0), $MachinePrecision] / N[(N[Exp[wj], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-26], N[(wj * N[(x * N[(wj * N[(N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] + N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(wj * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x / N[(N[Exp[wj], $MachinePrecision] * N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 4.0000000000000002e-26

   1. Initial program 64.4%

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

   3. Taylor expanded in wj around 0

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{fma}\left(wj, \color{blue}{x \cdot \left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \frac{wj \cdot \left(1 - wj\right)}{x}\right) - 2\right)}, x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \color{blue}{\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \frac{wj \cdot \left(1 - wj\right)}{x}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}, x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\color{blue}{\left(\frac{wj \cdot \left(1 - wj\right)}{x} + wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(\color{blue}{wj \cdot \frac{1 - wj}{x}} + wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(wj \cdot \frac{1 - wj}{x} + wj \cdot \color{blue}{\left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)}\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(wj \cdot \frac{1 - wj}{x} + wj \cdot \left(\frac{5}{2} + \color{blue}{\frac{-8}{3}} \cdot wj\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\color{blue}{wj \cdot \left(\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(wj \cdot \left(\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right) + \color{blue}{-2}\right), x\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \color{blue}{\mathsf{fma}\left(wj, \frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right)}, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \color{blue}{\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)}, -2\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \color{blue}{\frac{1 - wj}{x}} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{\color{blue}{1 - wj}}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \color{blue}{\left(\frac{-8}{3} \cdot wj + \frac{5}{2}\right)}, -2\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \left(\color{blue}{wj \cdot \frac{-8}{3}} + \frac{5}{2}\right), -2\right), x\right) \]

      15. accelerator-lowering-fma.f64 98.7

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

   8. Simplified 98.7%

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

   if 4.0000000000000002e-26 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 93.8%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. associate-/l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. *-inverses N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      11. distribute-neg-frac2 N/A

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

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

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

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

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

      14. distribute-rgt1-in N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. +-lowering-+.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(wj, \frac{1}{wj + 1}, \frac{x}{e^{wj} \cdot \left(-1 - wj\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00086:\\ \;\;\;\;\mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00086)
   (fma
    wj
    (* x (fma wj (+ (/ (- 1.0 wj) x) (fma wj -2.6666666666666665 2.5)) -2.0))
    x)
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00086) {
		tmp = fma(wj, (x * fma(wj, (((1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	} else {
		tmp = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00086)
		tmp = fma(wj, Float64(x * fma(wj, Float64(Float64(Float64(1.0 - wj) / x) + fma(wj, -2.6666666666666665, 2.5)), -2.0)), x);
	else
		tmp = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00086], N[(wj * N[(x * N[(wj * N[(N[(N[(1.0 - wj), $MachinePrecision] / x), $MachinePrecision] + N[(wj * -2.6666666666666665 + 2.5), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[((-wj) * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 8.59999999999999979e-4

   1. Initial program 75.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 97.0%

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

   6. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{fma}\left(wj, \color{blue}{x \cdot \left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \frac{wj \cdot \left(1 - wj\right)}{x}\right) - 2\right)}, x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \color{blue}{\left(\left(wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right) + \frac{wj \cdot \left(1 - wj\right)}{x}\right) + \left(\mathsf{neg}\left(2\right)\right)\right)}, x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\color{blue}{\left(\frac{wj \cdot \left(1 - wj\right)}{x} + wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(\color{blue}{wj \cdot \frac{1 - wj}{x}} + wj \cdot \left(\frac{5}{2} - \frac{8}{3} \cdot wj\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(wj \cdot \frac{1 - wj}{x} + wj \cdot \color{blue}{\left(\frac{5}{2} + \left(\mathsf{neg}\left(\frac{8}{3}\right)\right) \cdot wj\right)}\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\left(wj \cdot \frac{1 - wj}{x} + wj \cdot \left(\frac{5}{2} + \color{blue}{\frac{-8}{3}} \cdot wj\right)\right) + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(\color{blue}{wj \cdot \left(\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right)} + \left(\mathsf{neg}\left(2\right)\right)\right), x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \left(wj \cdot \left(\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)\right) + \color{blue}{-2}\right), x\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(wj, x \cdot \color{blue}{\mathsf{fma}\left(wj, \frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right)}, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \color{blue}{\frac{1 - wj}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right)}, -2\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \color{blue}{\frac{1 - wj}{x}} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{\color{blue}{1 - wj}}{x} + \left(\frac{5}{2} + \frac{-8}{3} \cdot wj\right), -2\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \color{blue}{\left(\frac{-8}{3} \cdot wj + \frac{5}{2}\right)}, -2\right), x\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \left(\color{blue}{wj \cdot \frac{-8}{3}} + \frac{5}{2}\right), -2\right), x\right) \]

      15. accelerator-lowering-fma.f64 97.3

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

   8. Simplified 97.3%

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

   if 8.59999999999999979e-4 < wj

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 96.9

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

   5. Simplified 96.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. div-inv N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 97.4

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

   7. Applied egg-rr 97.4%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00086:\\ \;\;\;\;\mathsf{fma}\left(wj, x \cdot \mathsf{fma}\left(wj, \frac{1 - wj}{x} + \mathsf{fma}\left(wj, -2.6666666666666665, 2.5\right), -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.00038:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.00038)
   (fma wj (- wj (* wj wj)) x)
   (fma (- wj) (/ 1.0 (+ wj 1.0)) wj)))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.00038) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = fma(-wj, (1.0 / (wj + 1.0)), wj);
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.00038)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = fma(Float64(-wj), Float64(1.0 / Float64(wj + 1.0)), wj);
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.00038], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[((-wj) * N[(1.0 / N[(wj + 1.0), $MachinePrecision]), $MachinePrecision] + wj), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 3.8000000000000002e-4

   1. Initial program 75.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 97.0%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. neg-mul-1 N/A

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

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

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

      7. unpow2 N/A

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

      8. unsub-neg N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 96.8

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

   8. Simplified 96.8%

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

   if 3.8000000000000002e-4 < wj

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 96.9

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

   5. Simplified 96.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. div-inv N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 97.4

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

   7. Applied egg-rr 97.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.00038:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-wj, \frac{1}{wj + 1}, wj\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;wj \leq 0.000165:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= wj 0.000165) (fma wj (- wj (* wj wj)) x) (+ wj (/ wj (- -1.0 wj)))))

C

double code(double wj, double x) {
	double tmp;
	if (wj <= 0.000165) {
		tmp = fma(wj, (wj - (wj * wj)), x);
	} else {
		tmp = wj + (wj / (-1.0 - wj));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (wj <= 0.000165)
		tmp = fma(wj, Float64(wj - Float64(wj * wj)), x);
	else
		tmp = Float64(wj + Float64(wj / Float64(-1.0 - wj)));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[wj, 0.000165], N[(wj * N[(wj - N[(wj * wj), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj + N[(wj / N[(-1.0 - wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if wj < 1.65e-4

   1. Initial program 75.9%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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. Simplified 97.0%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. neg-mul-1 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. neg-mul-1 N/A

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

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

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

      7. unpow2 N/A

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

      8. unsub-neg N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 96.8

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

   8. Simplified 96.8%

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

   if 1.65e-4 < wj

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto wj - \color{blue}{\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}}} \]
   4. Step-by-step derivation

      1. distribute-rgt1-in N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. *-inverses N/A

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

      5. associate-*l/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 96.9

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

   5. Simplified 96.9%

      \[\leadsto wj - \color{blue}{\frac{wj}{wj + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \leq 0.000165:\\ \;\;\;\;\mathsf{fma}\left(wj, wj - wj \cdot wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj + \frac{wj}{-1 - wj}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.3% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. Simplified 95.1%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. neg-mul-1 N/A

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

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

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

   7. unpow2 N/A

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

   8. unsub-neg N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 94.9

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

8. Simplified 94.9%

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

Alternative 8: 95.0% accurate, 47.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(wj, wj, x)
end

Wolfram

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

Derivation

1. Initial program 74.8%

   \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
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. Simplified 95.1%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. neg-mul-1 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. neg-mul-1 N/A

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

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

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

   7. unpow2 N/A

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

   8. unsub-neg N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 94.9

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

8. Simplified 94.9%

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

9. Taylor expanded in wj around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. accelerator-lowering-fma.f64 94.6

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

11. Simplified 94.6%

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

Alternative 9: 83.8% accurate, 331.0× 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 74.8%

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

3. Taylor expanded in wj around 0

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

5. Simplified 80.5%

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

Alternative 10: 4.5% accurate, 331.0× 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 74.8%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.3%

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

Alternative 11: 3.3% 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 74.8%

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

3. Taylor expanded in wj around inf

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

5. Simplified 4.1%

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

6. Taylor expanded in wj around 0

   \[\leadsto \color{blue}{-1} \]
7. Step-by-step derivation

8. Simplified 3.0%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer Target 1: 79.0% 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 2024205 
(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))))))