Average Error: 0.3 → 0.3
Time: 1.2m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\log \left(\sqrt[3]{z}\right) + \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(\log \left(\sqrt[3]{z}\right) + \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right)
double f(double x, double y, double z, double t, double a) {
        double r2307525 = x;
        double r2307526 = y;
        double r2307527 = r2307525 + r2307526;
        double r2307528 = log(r2307527);
        double r2307529 = z;
        double r2307530 = log(r2307529);
        double r2307531 = r2307528 + r2307530;
        double r2307532 = t;
        double r2307533 = r2307531 - r2307532;
        double r2307534 = a;
        double r2307535 = 0.5;
        double r2307536 = r2307534 - r2307535;
        double r2307537 = log(r2307532);
        double r2307538 = r2307536 * r2307537;
        double r2307539 = r2307533 + r2307538;
        return r2307539;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2307540 = z;
        double r2307541 = cbrt(r2307540);
        double r2307542 = log(r2307541);
        double r2307543 = 2.0;
        double r2307544 = x;
        double r2307545 = y;
        double r2307546 = r2307544 + r2307545;
        double r2307547 = log(r2307546);
        double r2307548 = fma(r2307543, r2307542, r2307547);
        double r2307549 = r2307542 + r2307548;
        double r2307550 = t;
        double r2307551 = r2307549 - r2307550;
        double r2307552 = log(r2307550);
        double r2307553 = a;
        double r2307554 = 0.5;
        double r2307555 = r2307553 - r2307554;
        double r2307556 = r2307552 * r2307555;
        double r2307557 = r2307551 + r2307556;
        return r2307557;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.3

    \[\leadsto \left(\left(\log \left(x + y\right) + \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  4. Applied log-prod0.3

    \[\leadsto \left(\left(\log \left(x + y\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  5. Applied associate-+r+0.3

    \[\leadsto \left(\color{blue}{\left(\left(\log \left(x + y\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \log t\]

  6. Simplified0.3

    \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  7. Final simplification0.3

    \[\leadsto \left(\left(\log \left(\sqrt[3]{z}\right) + \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))