Average Error: 0.1 → 0.1
Time: 6.4s
Precision: 64
\[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
↓
\[\left(\mathsf{fma}\left(z, 1 - \log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), x + y\right) - z \cdot \log \left({t}^{\frac{1}{3}}\right)\right) + \left(a - 0.5\right) \cdot b\]
\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 r352669 = x;
        double r352670 = y;
        double r352671 = r352669 + r352670;
        double r352672 = z;
        double r352673 = r352671 + r352672;
        double r352674 = t;
        double r352675 = log(r352674);
        double r352676 = r352672 * r352675;
        double r352677 = r352673 - r352676;
        double r352678 = a;
        double r352679 = 0.5;
        double r352680 = r352678 - r352679;
        double r352681 = b;
        double r352682 = r352680 * r352681;
        double r352683 = r352677 + r352682;
        return r352683;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r352684 = z;
        double r352685 = 1.0;
        double r352686 = t;
        double r352687 = cbrt(r352686);
        double r352688 = r352687 * r352687;
        double r352689 = log(r352688);
        double r352690 = r352685 - r352689;
        double r352691 = x;
        double r352692 = y;
        double r352693 = r352691 + r352692;
        double r352694 = fma(r352684, r352690, r352693);
        double r352695 = 0.3333333333333333;
        double r352696 = pow(r352686, r352695);
        double r352697 = log(r352696);
        double r352698 = r352684 * r352697;
        double r352699 = r352694 - r352698;
        double r352700 = a;
        double r352701 = 0.5;
        double r352702 = r352700 - r352701;
        double r352703 = b;
        double r352704 = r352702 * r352703;
        double r352705 = r352699 + r352704;
        return r352705;
}

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 2020062 +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)))