Average Error: 0.7 → 0.7
Time: 8.6s
Precision: 64
\[1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}\]
↓
\[1 - \frac{\frac{1}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}{\sqrt[3]{\sqrt[3]{y - z}} \cdot \sqrt[3]{\sqrt[3]{y - z}}} \cdot \frac{\frac{\sqrt[3]{x}}{y - t}}{\sqrt[3]{\sqrt[3]{y - z}}}\]
1 - \frac{x}{\left(y - z\right) \cdot \left(y - t\right)}
↓
1 - \frac{\frac{1}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}{\sqrt[3]{\sqrt[3]{y - z}} \cdot \sqrt[3]{\sqrt[3]{y - z}}} \cdot \frac{\frac{\sqrt[3]{x}}{y - t}}{\sqrt[3]{\sqrt[3]{y - z}}}
double f(double x, double y, double z, double t) {
        double r240519 = 1.0;
        double r240520 = x;
        double r240521 = y;
        double r240522 = z;
        double r240523 = r240521 - r240522;
        double r240524 = t;
        double r240525 = r240521 - r240524;
        double r240526 = r240523 * r240525;
        double r240527 = r240520 / r240526;
        double r240528 = r240519 - r240527;
        return r240528;
}
↓
double f(double x, double y, double z, double t) {
        double r240529 = 1.0;
        double r240530 = 1.0;
        double r240531 = y;
        double r240532 = z;
        double r240533 = r240531 - r240532;
        double r240534 = cbrt(r240533);
        double r240535 = r240534 * r240534;
        double r240536 = r240530 / r240535;
        double r240537 = x;
        double r240538 = cbrt(r240537);
        double r240539 = r240538 * r240538;
        double r240540 = r240536 * r240539;
        double r240541 = cbrt(r240534);
        double r240542 = r240541 * r240541;
        double r240543 = r240540 / r240542;
        double r240544 = t;
        double r240545 = r240531 - r240544;
        double r240546 = r240538 / r240545;
        double r240547 = r240546 / r240541;
        double r240548 = r240543 * r240547;
        double r240549 = r240529 - r240548;
        return r240549;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Derivation

  1. Initial program 0.7

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

  3. Applied *-un-lft-identity0.7

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

  4. Applied times-frac1.1

    \[\leadsto 1 - \color{blue}{\frac{1}{y - z} \cdot \frac{x}{y - t}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt1.3

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

  7. Applied *-un-lft-identity1.3

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

  8. Applied times-frac1.3

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

  9. Applied associate-*l*1.3

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

  10. Simplified1.3

    \[\leadsto 1 - \frac{1}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \color{blue}{\frac{\frac{x}{y - t}}{\sqrt[3]{y - z}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt1.4

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

  13. Applied *-un-lft-identity1.4

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

  14. Applied add-cube-cbrt1.4

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

  15. Applied times-frac1.4

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

  16. Applied times-frac0.9

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

  17. Applied associate-*r*0.7

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

  18. Simplified0.7

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

  19. Final simplification0.7

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

Reproduce

herbie shell --seed 2020062 +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)))))