Average Error: 0.6 → 0.7
Time: 6.9s
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 r291047 = 1.0;
        double r291048 = x;
        double r291049 = y;
        double r291050 = z;
        double r291051 = r291049 - r291050;
        double r291052 = t;
        double r291053 = r291049 - r291052;
        double r291054 = r291051 * r291053;
        double r291055 = r291048 / r291054;
        double r291056 = r291047 - r291055;
        return r291056;
}
↓
double f(double x, double y, double z, double t) {
        double r291057 = 1.0;
        double r291058 = x;
        double r291059 = cbrt(r291058);
        double r291060 = r291059 * r291059;
        double r291061 = y;
        double r291062 = z;
        double r291063 = r291061 - r291062;
        double r291064 = r291060 / r291063;
        double r291065 = cbrt(r291061);
        double r291066 = r291065 * r291065;
        double r291067 = t;
        double r291068 = 1.0;
        double r291069 = r291067 * r291068;
        double r291070 = -r291069;
        double r291071 = fma(r291066, r291065, r291070);
        double r291072 = -r291067;
        double r291073 = fma(r291072, r291068, r291067);
        double r291074 = r291071 + r291073;
        double r291075 = r291059 / r291074;
        double r291076 = r291064 * r291075;
        double r291077 = r291057 - r291076;
        return r291077;
}

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)))))