Jmat.Real.lambertw, newton loop step

Average Error: 13.6 → 1.2

Time: 6.8s

Precision: 64

• could not determine a ground truth for program body

  1. wj = -2.086802225316269e+96
  2. x = -1.0554065328349116e+156

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 2.32999755055536264 \cdot 10^{-7}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-wj, \frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}} - wj, \mathsf{fma}\left(\frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}}, \frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}}, -{wj}^{2}\right) \cdot \left(wj + 1\right)\right)}{\left(wj + 1\right) \cdot \left(\frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}} + \left(-wj\right)\right)}\\ \end{array}\]

C

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

↓

double code(double wj, double x) {
	double temp;
	if ((wj <= 2.3299975505553626e-07)) {
		temp = ((x + pow(wj, 2.0)) - (2.0 * (wj * x)));
	} else {
		temp = (fma(-wj, ((((x / (pow(wj, 3.0) + 1.0)) * (fma(wj, wj, 1.0) - wj)) / exp(wj)) - wj), (fma((((x / (pow(wj, 3.0) + 1.0)) * (fma(wj, wj, 1.0) - wj)) / exp(wj)), (((x / (pow(wj, 3.0) + 1.0)) * (fma(wj, wj, 1.0) - wj)) / exp(wj)), -pow(wj, 2.0)) * (wj + 1.0))) / ((wj + 1.0) * ((((x / (pow(wj, 3.0) + 1.0)) * (fma(wj, wj, 1.0) - wj)) / exp(wj)) + -wj)));
	}
	return temp;
}

Error

Bits error versus wj

Bits error versus x

Target

Original	13.6
Target	13.0
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if wj < 2.3299975505553626e-07

   1. Initial program 13.3

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

   2. Simplified 13.3

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

   3. Taylor expanded around 0 0.9

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

   if 2.3299975505553626e-07 < wj

   1. Initial program 25.7

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

   2. Simplified 2.2

      \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 2.2

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

   5. Applied flip3-+ 2.2

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

   6. Applied associate-/r/ 2.2

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

   7. Applied times-frac 2.2

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

   8. Simplified 2.2

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

   9. Simplified 2.2

      \[\leadsto \left(\frac{x}{{wj}^{3} + 1} \cdot \color{blue}{\frac{wj \cdot wj + \left(1 - wj \cdot 1\right)}{e^{wj}}} + wj\right) - \frac{wj}{wj + 1}\]
   10. Using strategy rm

   11. Applied flip-+ 13.1

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

   12. Applied frac-sub 13.1

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

   13. Simplified 13.1

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

   14. Simplified 13.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 2.32999755055536264 \cdot 10^{-7}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-wj, \frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}} - wj, \mathsf{fma}\left(\frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}}, \frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}}, -{wj}^{2}\right) \cdot \left(wj + 1\right)\right)}{\left(wj + 1\right) \cdot \left(\frac{\frac{x}{{wj}^{3} + 1} \cdot \left(\mathsf{fma}\left(wj, wj, 1\right) - wj\right)}{e^{wj}} + \left(-wj\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))