Jmat.Real.lambertw, newton loop step

Percentage Accurate: 78.7% → 99.6%

Time: 9.1s

Alternatives: 15

Speedup: 55.2×

• could not determine a ground truth

  1. wj = 5.112546658414614e+59
  2. x = -1.378275235561124e-283

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: 78.7% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-wj}\\ t_1 := e^{wj} \cdot wj\\ t_2 := \frac{x}{wj - -1}\\ \mathbf{if}\;wj - \frac{t\_1 - x}{t\_1 + e^{wj}} \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t\_0, \left(\left(1 - wj\right) \cdot wj\right) \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t\_0, wj - \frac{wj}{wj - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (let* ((t_0 (exp (- wj))) (t_1 (* (exp wj) wj)) (t_2 (/ x (- wj -1.0))))
   (if (<= (- wj (/ (- t_1 x) (+ t_1 (exp wj)))) 1e-15)
     (fma t_2 t_0 (* (* (- 1.0 wj) wj) wj))
     (fma t_2 t_0 (- wj (/ wj (- wj -1.0)))))))

C

double code(double wj, double x) {
	double t_0 = exp(-wj);
	double t_1 = exp(wj) * wj;
	double t_2 = x / (wj - -1.0);
	double tmp;
	if ((wj - ((t_1 - x) / (t_1 + exp(wj)))) <= 1e-15) {
		tmp = fma(t_2, t_0, (((1.0 - wj) * wj) * wj));
	} else {
		tmp = fma(t_2, t_0, (wj - (wj / (wj - -1.0))));
	}
	return tmp;
}

Julia

function code(wj, x)
	t_0 = exp(Float64(-wj))
	t_1 = Float64(exp(wj) * wj)
	t_2 = Float64(x / Float64(wj - -1.0))
	tmp = 0.0
	if (Float64(wj - Float64(Float64(t_1 - x) / Float64(t_1 + exp(wj)))) <= 1e-15)
		tmp = fma(t_2, t_0, Float64(Float64(Float64(1.0 - wj) * wj) * wj));
	else
		tmp = fma(t_2, t_0, Float64(wj - Float64(wj / Float64(wj - -1.0))));
	end
	return tmp
end

Wolfram

code[wj_, x_] := Block[{t$95$0 = N[Exp[(-wj)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[wj], $MachinePrecision] * wj), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(wj - N[(N[(t$95$1 - x), $MachinePrecision] / N[(t$95$1 + N[Exp[wj], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-15], N[(t$95$2 * t$95$0 + N[(N[(N[(1.0 - wj), $MachinePrecision] * wj), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * t$95$0 + N[(wj - 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))))) < 1.0000000000000001e-15

   1. Initial program 77.3%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 98.1%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. *-rgt-identity N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. times-frac N/A

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

       7. sub-neg N/A

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

       8. mul-1-neg N/A

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

       9. lower-fma.f64 N/A

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

   11. Applied rewrites 89.1%

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

   12. Taylor expanded in wj around 0

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

   14. Applied rewrites 99.7%

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

   if 1.0000000000000001e-15 < (-.f64 wj (/.f64 (-.f64 (*.f64 wj (exp.f64 wj)) x) (+.f64 (exp.f64 wj) (*.f64 wj (exp.f64 wj)))))

   1. Initial program 94.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 90.4%

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

   10. Applied rewrites 90.4%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. *-rgt-identity N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. times-frac N/A

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

       7. exp-neg N/A

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

       8. sub-neg N/A

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

       9. distribute-rgt1-in N/A

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

       10. +-commutative N/A

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

       11. times-frac N/A

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

       12. *-inverses N/A

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

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

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

       14. metadata-eval N/A

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

   13. Applied rewrites 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{e^{wj} \cdot wj - x}{e^{wj} \cdot wj + e^{wj}} \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, \left(\left(1 - wj\right) \cdot wj\right) \cdot wj\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, wj - \frac{wj}{wj - -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. *-rgt-identity N/A

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

    4. distribute-rgt1-in N/A

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

    5. +-commutative N/A

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

    6. times-frac N/A

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

    7. sub-neg N/A

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

    8. mul-1-neg N/A

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

    9. lower-fma.f64 N/A

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

11. Applied rewrites 92.0%

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

12. Taylor expanded in wj around 0

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

14. Applied rewrites 98.4%

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

15. Final simplification 98.4%

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

Alternative 3: 98.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. *-rgt-identity N/A

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

    4. distribute-rgt1-in N/A

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

    5. +-commutative N/A

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

    6. times-frac N/A

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

    7. sub-neg N/A

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

    8. mul-1-neg N/A

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

    9. lower-fma.f64 N/A

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

11. Applied rewrites 92.0%

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

12. Taylor expanded in wj around 0

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

14. Applied rewrites 98.4%

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

15. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. *-rgt-identity N/A

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

    4. distribute-rgt1-in N/A

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

    5. +-commutative N/A

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

    6. times-frac N/A

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

    7. sub-neg N/A

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

    8. mul-1-neg N/A

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

    9. lower-fma.f64 N/A

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

11. Applied rewrites 92.0%

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

12. Taylor expanded in wj around 0

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

14. Applied rewrites 97.8%

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

15. Final simplification 97.8%

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

Alternative 5: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, wj - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (if (<= x 5e+17)
   (fma
    (fma
     (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
     wj
     (* -2.0 x))
    wj
    x)
   (fma (/ x (- wj -1.0)) (exp (- wj)) (- wj 1.0))))

C

double code(double wj, double x) {
	double tmp;
	if (x <= 5e+17) {
		tmp = fma(fma(fma(2.5, x, (1.0 - (fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, (-2.0 * x)), wj, x);
	} else {
		tmp = fma((x / (wj - -1.0)), exp(-wj), (wj - 1.0));
	}
	return tmp;
}

Julia

function code(wj, x)
	tmp = 0.0
	if (x <= 5e+17)
		tmp = fma(fma(fma(2.5, x, Float64(1.0 - Float64(fma(0.6666666666666666, x, fma(2.0, x, 1.0)) * wj))), wj, Float64(-2.0 * x)), wj, x);
	else
		tmp = fma(Float64(x / Float64(wj - -1.0)), exp(Float64(-wj)), Float64(wj - 1.0));
	end
	return tmp
end

Wolfram

code[wj_, x_] := If[LessEqual[x, 5e+17], N[(N[(N[(2.5 * x + N[(1.0 - N[(N[(0.6666666666666666 * x + N[(2.0 * x + 1.0), $MachinePrecision]), $MachinePrecision] * wj), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * wj + N[(-2.0 * x), $MachinePrecision]), $MachinePrecision] * wj + x), $MachinePrecision], N[(N[(x / N[(wj - -1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[(-wj)], $MachinePrecision] + N[(wj - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e17

   1. Initial program 77.6%

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \]
   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 97.1%

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

   if 5e17 < x

   1. Initial program 96.9%

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

   3. Taylor expanded in wj around 0

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in wj around 0

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

   8. Applied rewrites 93.3%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate--l+ N/A

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

       3. *-rgt-identity N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. times-frac N/A

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

       7. sub-neg N/A

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

       8. mul-1-neg N/A

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

       9. lower-fma.f64 N/A

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in wj around inf

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

   14. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{wj - -1}, e^{-wj}, wj - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate--l+ N/A

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

    3. *-rgt-identity N/A

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

    4. distribute-rgt1-in N/A

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

    5. +-commutative N/A

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

    6. times-frac N/A

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

    7. sub-neg N/A

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

    8. mul-1-neg N/A

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

    9. lower-fma.f64 N/A

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

11. Applied rewrites 92.0%

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

12. Taylor expanded in wj around 0

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

14. Applied rewrites 97.2%

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

15. Final simplification 97.2%

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

Alternative 7: 96.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.5, x, 1 - \mathsf{fma}\left(0.6666666666666666, x, \mathsf{fma}\left(2, x, 1\right)\right) \cdot wj\right), wj, -2 \cdot x\right), wj, x\right) \end{array} \]

FPCore

(FPCore (wj x)
 :precision binary64
 (fma
  (fma
   (fma 2.5 x (- 1.0 (* (fma 0.6666666666666666 x (fma 2.0 x 1.0)) wj)))
   wj
   (* -2.0 x))
  wj
  x))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

Alternative 8: 96.0% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(wj, x)
	return fma(fma(fma(wj, 2.5, -2.0), x, wj), wj, x)
end

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 95.3%

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

Alternative 9: 95.7% accurate, 22.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.0%

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

Alternative 10: 95.8% accurate, 25.5× speedup

Math

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

FPCore

(FPCore (wj x) :precision binary64 (fma (fma -2.0 x wj) wj x))

C

double code(double wj, double x) {
	return fma(fma(-2.0, x, wj), wj, x);
}

Julia

function code(wj, x)
	return fma(fma(-2.0, x, wj), wj, x)
end

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

8. Applied rewrites 95.3%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 94.9%

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

Alternative 11: 85.4% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 86.3

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

5. Applied rewrites 86.3%

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

6. Final simplification 86.3%

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

Alternative 12: 85.4% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in wj around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. +-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 86.3

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

8. Applied rewrites 86.3%

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

Alternative 13: 84.8% accurate, 55.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 95.3

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

5. Applied rewrites 95.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 86.7%

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

9. Taylor expanded in wj around 0

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

11. Applied rewrites 85.7%

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

Alternative 14: 13.4% accurate, 55.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. *-commutative N/A

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

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

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 95.3

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

5. Applied rewrites 95.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 11.6%

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

Alternative 15: 4.1% accurate, 82.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in wj around inf

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

5. Applied rewrites 4.4%

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

Developer Target 1: 79.6% 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 2024249 
(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))))))