Average Error: 0.3 → 0.3
Time: 32.6s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{y + x}\right) + \left(\log z - t\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{y + x}\right) + \left(\log z - t\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r15877381 = x;
        double r15877382 = y;
        double r15877383 = r15877381 + r15877382;
        double r15877384 = log(r15877383);
        double r15877385 = z;
        double r15877386 = log(r15877385);
        double r15877387 = r15877384 + r15877386;
        double r15877388 = t;
        double r15877389 = r15877387 - r15877388;
        double r15877390 = a;
        double r15877391 = 0.5;
        double r15877392 = r15877390 - r15877391;
        double r15877393 = log(r15877388);
        double r15877394 = r15877392 * r15877393;
        double r15877395 = r15877389 + r15877394;
        return r15877395;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r15877396 = t;
        double r15877397 = log(r15877396);
        double r15877398 = a;
        double r15877399 = 0.5;
        double r15877400 = r15877398 - r15877399;
        double r15877401 = y;
        double r15877402 = x;
        double r15877403 = r15877401 + r15877402;
        double r15877404 = cbrt(r15877403);
        double r15877405 = log(r15877404);
        double r15877406 = z;
        double r15877407 = log(r15877406);
        double r15877408 = r15877407 - r15877396;
        double r15877409 = r15877405 + r15877408;
        double r15877410 = r15877404 * r15877404;
        double r15877411 = log(r15877410);
        double r15877412 = r15877409 + r15877411;
        double r15877413 = fma(r15877397, r15877400, r15877412);
        return r15877413;
}

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. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - 0.5, \log \left(y + x\right) + \left(\log z - t\right)\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +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))))