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 r16246649 = x;
        double r16246650 = y;
        double r16246651 = r16246649 + r16246650;
        double r16246652 = log(r16246651);
        double r16246653 = z;
        double r16246654 = log(r16246653);
        double r16246655 = r16246652 + r16246654;
        double r16246656 = t;
        double r16246657 = r16246655 - r16246656;
        double r16246658 = a;
        double r16246659 = 0.5;
        double r16246660 = r16246658 - r16246659;
        double r16246661 = log(r16246656);
        double r16246662 = r16246660 * r16246661;
        double r16246663 = r16246657 + r16246662;
        return r16246663;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r16246664 = z;
        double r16246665 = cbrt(r16246664);
        double r16246666 = log(r16246665);
        double r16246667 = 2.0;
        double r16246668 = x;
        double r16246669 = y;
        double r16246670 = r16246668 + r16246669;
        double r16246671 = log(r16246670);
        double r16246672 = fma(r16246667, r16246666, r16246671);
        double r16246673 = r16246666 + r16246672;
        double r16246674 = t;
        double r16246675 = r16246673 - r16246674;
        double r16246676 = log(r16246674);
        double r16246677 = a;
        double r16246678 = 0.5;
        double r16246679 = r16246677 - r16246678;
        double r16246680 = r16246676 * r16246679;
        double r16246681 = r16246675 + r16246680;
        return r16246681;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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"

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))