Average Error: 0.1 → 0.1
Time: 8.4s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\left(\mathsf{fma}\left(z, 1 - \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), x + y\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]
\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[3]{t} \cdot \sqrt[3]{t}\right), x + y\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b
double f(double x, double y, double z, double t, double a, double b) {
        double r1032681 = x;
        double r1032682 = y;
        double r1032683 = r1032681 + r1032682;
        double r1032684 = z;
        double r1032685 = r1032683 + r1032684;
        double r1032686 = t;
        double r1032687 = log(r1032686);
        double r1032688 = r1032684 * r1032687;
        double r1032689 = r1032685 - r1032688;
        double r1032690 = a;
        double r1032691 = 0.5;
        double r1032692 = r1032690 - r1032691;
        double r1032693 = b;
        double r1032694 = r1032692 * r1032693;
        double r1032695 = r1032689 + r1032694;
        return r1032695;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1032696 = z;
        double r1032697 = 1.0;
        double r1032698 = t;
        double r1032699 = cbrt(r1032698);
        double r1032700 = r1032699 * r1032699;
        double r1032701 = log(r1032700);
        double r1032702 = r1032697 - r1032701;
        double r1032703 = x;
        double r1032704 = y;
        double r1032705 = r1032703 + r1032704;
        double r1032706 = fma(r1032696, r1032702, r1032705);
        double r1032707 = 0.3333333333333333;
        double r1032708 = pow(r1032698, r1032707);
        double r1032709 = log(r1032708);
        double r1032710 = r1032696 * r1032709;
        double r1032711 = r1032706 - r1032710;
        double r1032712 = a;
        double r1032713 = 0.5;
        double r1032714 = r1032712 - r1032713;
        double r1032715 = b;
        double r1032716 = r1032714 * r1032715;
        double r1032717 = r1032711 + r1032716;
        return r1032717;
}

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
Target0.3
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-cube-cbrt0.1

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

  4. Applied log-prod0.1

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

  5. Applied distribute-lft-in0.1

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

  6. Applied associate--r+0.1

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

  7. Simplified0.1

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

  9. Applied pow1/30.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020034 +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)))