Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.6% → 98.7%

Time: 11.3s

Alternatives: 11

Speedup: 27.6×

• could not determine a ground truth

  1. wj = 1.5731416034648811e+264
  2. x = -1901720352.4947522

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ \mathbf{if}\;wj + \frac{x - t\_0}{e^{wj} + t\_0} \leq 1.5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(x, \frac{-1}{e^{wj} \cdot \left(wj + 1\right)}, \frac{wj}{wj + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj))))
   (if (<= (+ wj (/ (- x t_0) (+ (exp wj) t_0))) 1.5e-13)
     (fma wj (fma wj (- 1.0 wj) (* x -2.0)) x)
     (- wj (fma x (/ -1.0 (* (exp wj) (+ wj 1.0))) (/ wj (+ wj 1.0)))))))

C

double code(double wj, double x) {
	double t_0 = wj * exp(wj);
	double tmp;
	if ((wj + ((x - t_0) / (exp(wj) + t_0))) <= 1.5e-13) {
		tmp = fma(wj, fma(wj, (1.0 - wj), (x * -2.0)), x);
	} else {
		tmp = wj - fma(x, (-1.0 / (exp(wj) * (wj + 1.0))), (wj / (wj + 1.0)));
	}
	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))) <= 1.5e-13)
		tmp = fma(wj, fma(wj, Float64(1.0 - wj), Float64(x * -2.0)), x);
	else
		tmp = Float64(wj - fma(x, Float64(-1.0 / Float64(exp(wj) * Float64(wj + 1.0))), Float64(wj / Float64(wj + 1.0))));
	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], 1.5e-13], N[(wj * N[(wj * N[(1.0 - wj), $MachinePrecision] + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(wj - N[(x * N[(-1.0 / N[(N[Exp[wj], $MachinePrecision] * N[(wj + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(wj / N[(wj + 1.0), $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))))) < 1.49999999999999992e-13

   1. Initial program 73.7%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.8%

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

   if 1.49999999999999992e-13 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 96.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

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

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

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

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

      10. distribute-frac-neg2 N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 1.5 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(wj, \mathsf{fma}\left(wj, 1 - wj, x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \mathsf{fma}\left(x, \frac{-1}{e^{wj} \cdot \left(wj + 1\right)}, \frac{wj}{wj + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := wj \cdot e^{wj}\\ t_1 := wj + \frac{x - t\_0}{e^{wj} + t\_0}\\ t_2 := wj - \left(-x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (* wj (exp wj)))
        (t_1 (+ wj (/ (- x t_0) (+ (exp wj) t_0))))
        (t_2 (- wj (- x))))
   (if (<= t_1 -5e-250) t_2 (if (<= t_1 0.0) (* wj wj) t_2))))

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 t_2 = wj - -x;
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = wj * exp(wj)
    t_1 = wj + ((x - t_0) / (exp(wj) + t_0))
    t_2 = wj - -x
    if (t_1 <= (-5d-250)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = wj * wj
    else
        tmp = t_2
    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 t_2 = wj - -x;
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = wj * wj;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(wj, x):
	t_0 = wj * math.exp(wj)
	t_1 = wj + ((x - t_0) / (math.exp(wj) + t_0))
	t_2 = wj - -x
	tmp = 0
	if t_1 <= -5e-250:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = wj * wj
	else:
		tmp = t_2
	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)))
	t_2 = Float64(wj - Float64(-x))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(wj * wj);
	else
		tmp = t_2;
	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));
	t_2 = wj - -x;
	tmp = 0.0;
	if (t_1 <= -5e-250)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = wj * wj;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(wj - (-x)), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-250], t$95$2, If[LessEqual[t$95$1, 0.0], N[(wj * wj), $MachinePrecision], t$95$2]]]]]

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))))) < -5.00000000000000027e-250 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 95.9%

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

   3. Taylor expanded in wj around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if -5.00000000000000027e-250 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj))))) < 0.0

   1. Initial program 7.9%

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

   3. Taylor expanded in wj around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq -5 \cdot 10^{-250}:\\ \;\;\;\;wj - \left(-x\right)\\ \mathbf{elif}\;wj + \frac{x - wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} \leq 0:\\ \;\;\;\;wj \cdot wj\\ \mathbf{else}:\\ \;\;\;\;wj - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 79.4% 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 2024226 
(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))))))