Average Error: 0.1 → 0.1
Time: 29.8s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\mathsf{fma}\left(z, \mathsf{fma}\left(-2, \log \left(\sqrt[3]{t}\right), 1\right) - \log \left(\sqrt[3]{t}\right), \mathsf{fma}\left(a - 0.5, b, x + y\right)\right)\]
\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b
↓
\mathsf{fma}\left(z, \mathsf{fma}\left(-2, \log \left(\sqrt[3]{t}\right), 1\right) - \log \left(\sqrt[3]{t}\right), \mathsf{fma}\left(a - 0.5, b, x + y\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r290345 = x;
        double r290346 = y;
        double r290347 = r290345 + r290346;
        double r290348 = z;
        double r290349 = r290347 + r290348;
        double r290350 = t;
        double r290351 = log(r290350);
        double r290352 = r290348 * r290351;
        double r290353 = r290349 - r290352;
        double r290354 = a;
        double r290355 = 0.5;
        double r290356 = r290354 - r290355;
        double r290357 = b;
        double r290358 = r290356 * r290357;
        double r290359 = r290353 + r290358;
        return r290359;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r290360 = z;
        double r290361 = -2.0;
        double r290362 = t;
        double r290363 = cbrt(r290362);
        double r290364 = log(r290363);
        double r290365 = 1.0;
        double r290366 = fma(r290361, r290364, r290365);
        double r290367 = r290366 - r290364;
        double r290368 = a;
        double r290369 = 0.5;
        double r290370 = r290368 - r290369;
        double r290371 = b;
        double r290372 = x;
        double r290373 = y;
        double r290374 = r290372 + r290373;
        double r290375 = fma(r290370, r290371, r290374);
        double r290376 = fma(r290360, r290367, r290375);
        return r290376;
}

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. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, 1 - \log t, \mathsf{fma}\left(a - 0.5, b, x + y\right)\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied associate--r+0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 +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)))