Average Error: 0.6 → 0.7
Time: 13.7s
Precision: 64
\[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\]
↓
\[1 - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{y - z} \cdot \frac{\sqrt[3]{x}}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -t \cdot 1\right) + \mathsf{fma}\left(-t, 1, t\right)}\]
1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}
↓
1 - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{y - z} \cdot \frac{\sqrt[3]{x}}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -t \cdot 1\right) + \mathsf{fma}\left(-t, 1, t\right)}
double f(double x, double y, double z, double t) {
        double r300986 = 1.0;
        double r300987 = x;
        double r300988 = y;
        double r300989 = z;
        double r300990 = r300988 - r300989;
        double r300991 = t;
        double r300992 = r300988 - r300991;
        double r300993 = r300990 * r300992;
        double r300994 = r300987 / r300993;
        double r300995 = r300986 - r300994;
        return r300995;
}
↓
double f(double x, double y, double z, double t) {
        double r300996 = 1.0;
        double r300997 = x;
        double r300998 = cbrt(r300997);
        double r300999 = r300998 * r300998;
        double r301000 = y;
        double r301001 = z;
        double r301002 = r301000 - r301001;
        double r301003 = r300999 / r301002;
        double r301004 = cbrt(r301000);
        double r301005 = r301004 * r301004;
        double r301006 = t;
        double r301007 = 1.0;
        double r301008 = r301006 * r301007;
        double r301009 = -r301008;
        double r301010 = fma(r301005, r301004, r301009);
        double r301011 = -r301006;
        double r301012 = fma(r301011, r301007, r301006);
        double r301013 = r301010 + r301012;
        double r301014 = r300998 / r301013;
        double r301015 = r301003 * r301014;
        double r301016 = r300996 - r301015;
        return r301016;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.6

    \[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.7

    \[\leadsto 1 - \frac{x}{\left(y - z\right) \cdot \left(y - \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}\]

  4. Applied add-cube-cbrt0.8

    \[\leadsto 1 - \frac{x}{\left(y - z\right) \cdot \left(\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}\]

  5. Applied prod-diff0.8

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

  6. Applied distribute-lft-in7.4

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

  7. Simplified7.3

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

  8. Simplified0.7

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

  10. Applied distribute-lft-out0.7

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

  11. Applied add-cube-cbrt0.8

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

  12. Applied times-frac0.7

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

  13. Final simplification0.7

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, A"
  :precision binary64
  (- 1 (/ x (* (- y z) (- y t)))))