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

Average Error: 15.8 → 0.3

Time: 19.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.435048613210959909118539310961078344525 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{{x}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r287483 = x;
        double r287484 = y;
        double r287485 = r287483 / r287484;
        double r287486 = log(r287485);
        double r287487 = r287483 * r287486;
        double r287488 = z;
        double r287489 = r287487 - r287488;
        return r287489;
}

↓

double f(double x, double y, double z) {
        double r287490 = y;
        double r287491 = -5.43504861321096e-309;
        bool r287492 = r287490 <= r287491;
        double r287493 = x;
        double r287494 = -1.0;
        double r287495 = r287494 / r287490;
        double r287496 = log(r287495);
        double r287497 = r287494 / r287493;
        double r287498 = log(r287497);
        double r287499 = r287496 - r287498;
        double r287500 = r287493 * r287499;
        double r287501 = z;
        double r287502 = r287500 - r287501;
        double r287503 = 2.0;
        double r287504 = cbrt(r287493);
        double r287505 = cbrt(r287490);
        double r287506 = r287504 / r287505;
        double r287507 = log(r287506);
        double r287508 = r287503 * r287507;
        double r287509 = r287493 * r287508;
        double r287510 = 0.6666666666666666;
        double r287511 = pow(r287493, r287510);
        double r287512 = cbrt(r287511);
        double r287513 = cbrt(r287504);
        double r287514 = r287512 * r287513;
        double r287515 = r287514 / r287505;
        double r287516 = log(r287515);
        double r287517 = r287493 * r287516;
        double r287518 = r287509 + r287517;
        double r287519 = r287518 - r287501;
        double r287520 = r287492 ? r287502 : r287519;
        return r287520;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.8
Target	8.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -5.43504861321096e-309

   1. Initial program 16.3

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

   2. Taylor expanded around -inf 0.3

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

   if -5.43504861321096e-309 < y

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.2

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

   4. Applied add-cube-cbrt 15.2

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

   5. Applied times-frac 15.2

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

   6. Applied log-prod 3.7

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

   7. Applied distribute-lft-in 3.7

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

   8. Simplified 0.2

      \[\leadsto \left(\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 0.2

       \[\leadsto \left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\sqrt[3]{y}}\right)\right) - z\]

   11. Applied cbrt-prod 0.2

       \[\leadsto \left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\frac{\color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{y}}\right)\right) - z\]

   12. Simplified 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.435048613210959909118539310961078344525 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{{x}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))