Average Error: 0.3 → 0.3
Time: 24.0s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(2 \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(2 \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)
double f(double x, double y, double z, double t, double a) {
        double r393406 = x;
        double r393407 = y;
        double r393408 = r393406 + r393407;
        double r393409 = log(r393408);
        double r393410 = z;
        double r393411 = log(r393410);
        double r393412 = r393409 + r393411;
        double r393413 = t;
        double r393414 = r393412 - r393413;
        double r393415 = a;
        double r393416 = 0.5;
        double r393417 = r393415 - r393416;
        double r393418 = log(r393413);
        double r393419 = r393417 * r393418;
        double r393420 = r393414 + r393419;
        return r393420;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r393421 = 2.0;
        double r393422 = 1.0;
        double r393423 = t;
        double r393424 = r393422 / r393423;
        double r393425 = -0.3333333333333333;
        double r393426 = pow(r393424, r393425);
        double r393427 = log(r393426);
        double r393428 = r393421 * r393427;
        double r393429 = a;
        double r393430 = 0.5;
        double r393431 = r393429 - r393430;
        double r393432 = x;
        double r393433 = y;
        double r393434 = r393432 + r393433;
        double r393435 = log(r393434);
        double r393436 = z;
        double r393437 = log(r393436);
        double r393438 = r393435 + r393437;
        double r393439 = r393438 - r393423;
        double r393440 = fma(r393428, r393431, r393439);
        double r393441 = cbrt(r393423);
        double r393442 = log(r393441);
        double r393443 = r393431 * r393442;
        double r393444 = r393440 + r393443;
        return r393444;
}

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

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  8. Taylor expanded around inf 0.3

    \[\leadsto \mathsf{fma}\left(2 \cdot \log \color{blue}{\left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)}, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\]

  9. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(2 \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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