Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 15.1 → 0.5

Time: 3.4s

Precision: 64

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.9380282042788273 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{x \cdot \left({\left(\log \left(\frac{-1}{y}\right)\right)}^{3} - {\left(\log \left(--1\right)\right)}^{3}\right)}{\log \left(\frac{-1}{y}\right) \cdot \log \left(\frac{-1}{y}\right) + \left(\log \left(--1\right) \cdot \log \left(--1\right) + \log \left(\frac{-1}{y}\right) \cdot \log \left(--1\right)\right)} + x \cdot \log \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot x + \left(-\mathsf{fma}\left(x, \log y, z\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((x * log((x / y))) - z);
}

↓

double code(double x, double y, double z) {
	double VAR;
	if ((y <= -5.938028204278827e-309)) {
		VAR = ((((x * (pow(log((-1.0 / y)), 3.0) - pow(log(--1.0), 3.0))) / ((log((-1.0 / y)) * log((-1.0 / y))) + ((log(--1.0) * log(--1.0)) + (log((-1.0 / y)) * log(--1.0))))) + (x * log(-x))) - z);
	} else {
		VAR = ((log(x) * x) + -fma(x, log(y), z));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.1
Target	7.6
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -5.938028204278827e-309

   1. Initial program 14.9

      \[x \cdot \log \left(\frac{x}{y}\right) - z\]

   2. Taylor expanded around -inf 0.4

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right)} - z\]
   3. Using strategy rm

   4. Applied frac-2neg 0.4

      \[\leadsto x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \color{blue}{\left(\frac{--1}{-x}\right)}\right) - z\]

   5. Applied log-div 0.3

      \[\leadsto x \cdot \left(\log \left(\frac{-1}{y}\right) - \color{blue}{\left(\log \left(--1\right) - \log \left(-x\right)\right)}\right) - z\]

   6. Applied associate--r- 0.3

      \[\leadsto x \cdot \color{blue}{\left(\left(\log \left(\frac{-1}{y}\right) - \log \left(--1\right)\right) + \log \left(-x\right)\right)} - z\]

   7. Applied distribute-lft-in 0.4

      \[\leadsto \color{blue}{\left(x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(--1\right)\right) + x \cdot \log \left(-x\right)\right)} - z\]
   8. Using strategy rm

   9. Applied flip3-- 0.4

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

   10. Applied associate-*r/ 0.7

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

   if -5.938028204278827e-309 < y

   1. Initial program 15.4

      \[x \cdot \log \left(\frac{x}{y}\right) - z\]
   2. Using strategy rm

   3. Applied div-inv 15.4

      \[\leadsto x \cdot \log \color{blue}{\left(x \cdot \frac{1}{y}\right)} - z\]

   4. Applied log-prod 0.3

      \[\leadsto x \cdot \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)} - z\]

   5. Applied distribute-rgt-in 0.4

      \[\leadsto \color{blue}{\left(\log x \cdot x + \log \left(\frac{1}{y}\right) \cdot x\right)} - z\]

   6. Applied associate--l+ 0.4

      \[\leadsto \color{blue}{\log x \cdot x + \left(\log \left(\frac{1}{y}\right) \cdot x - z\right)}\]

   7. Simplified 0.3

      \[\leadsto \log x \cdot x + \color{blue}{\left(-\mathsf{fma}\left(x, \log y, z\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.9380282042788273 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{x \cdot \left({\left(\log \left(\frac{-1}{y}\right)\right)}^{3} - {\left(\log \left(--1\right)\right)}^{3}\right)}{\log \left(\frac{-1}{y}\right) \cdot \log \left(\frac{-1}{y}\right) + \left(\log \left(--1\right) \cdot \log \left(--1\right) + \log \left(\frac{-1}{y}\right) \cdot \log \left(--1\right)\right)} + x \cdot \log \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot x + \left(-\mathsf{fma}\left(x, \log y, z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))