Average Error: 0.1 → 0.1
Time: 6.4s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\left(\mathsf{fma}\left(z, 1 - \log \left(\sqrt{t}\right), x + y\right) - z \cdot \log \left(\sqrt{t}\right)\right) + \left(a - 0.5\right) \cdot b\]
\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\left(\mathsf{fma}\left(z, 1 - \log \left(\sqrt{t}\right), x + y\right) - z \cdot \log \left(\sqrt{t}\right)\right) + \left(a - 0.5\right) \cdot b
double f(double x, double y, double z, double t, double a, double b) {
        double r368348 = x;
        double r368349 = y;
        double r368350 = r368348 + r368349;
        double r368351 = z;
        double r368352 = r368350 + r368351;
        double r368353 = t;
        double r368354 = log(r368353);
        double r368355 = r368351 * r368354;
        double r368356 = r368352 - r368355;
        double r368357 = a;
        double r368358 = 0.5;
        double r368359 = r368357 - r368358;
        double r368360 = b;
        double r368361 = r368359 * r368360;
        double r368362 = r368356 + r368361;
        return r368362;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r368363 = z;
        double r368364 = 1.0;
        double r368365 = t;
        double r368366 = sqrt(r368365);
        double r368367 = log(r368366);
        double r368368 = r368364 - r368367;
        double r368369 = x;
        double r368370 = y;
        double r368371 = r368369 + r368370;
        double r368372 = fma(r368363, r368368, r368371);
        double r368373 = r368363 * r368367;
        double r368374 = r368372 - r368373;
        double r368375 = a;
        double r368376 = 0.5;
        double r368377 = r368375 - r368376;
        double r368378 = b;
        double r368379 = r368377 * r368378;
        double r368380 = r368374 + r368379;
        return r368380;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original0.1
Target0.4
Herbie0.1
\[\left(\left(x + y\right) + \frac{\left(1 - {\left(\log t\right)}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b\]

Derivation

  1. Initial program 0.1

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

  3. Applied add-sqr-sqrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Applied associate--r+0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

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