Average Error: 9.5 → 0.1
Time: 2.4s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{z}}{t}, \frac{2}{t}\right) - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{z}}{t}, \frac{2}{t}\right) - 2\right)
double f(double x, double y, double z, double t) {
        double r946474 = x;
        double r946475 = y;
        double r946476 = r946474 / r946475;
        double r946477 = 2.0;
        double r946478 = z;
        double r946479 = r946478 * r946477;
        double r946480 = 1.0;
        double r946481 = t;
        double r946482 = r946480 - r946481;
        double r946483 = r946479 * r946482;
        double r946484 = r946477 + r946483;
        double r946485 = r946481 * r946478;
        double r946486 = r946484 / r946485;
        double r946487 = r946476 + r946486;
        return r946487;
}
↓
double f(double x, double y, double z, double t) {
        double r946488 = x;
        double r946489 = y;
        double r946490 = r946488 / r946489;
        double r946491 = 2.0;
        double r946492 = 1.0;
        double r946493 = z;
        double r946494 = r946492 / r946493;
        double r946495 = t;
        double r946496 = r946494 / r946495;
        double r946497 = r946491 / r946495;
        double r946498 = fma(r946491, r946496, r946497);
        double r946499 = r946498 - r946491;
        double r946500 = r946490 + r946499;
        return r946500;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.5
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.5

    \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t \cdot z} + 2 \cdot \frac{1}{t}\right) - 2\right)}\]

  3. Simplified0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\left(\mathsf{fma}\left(2, \frac{1}{t \cdot z}, \frac{2}{t}\right) - 2\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{t \cdot z}, \frac{2}{t}\right) - 2\right)\]

  6. Applied times-frac0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{t} \cdot \frac{\sqrt[3]{1}}{z}}, \frac{2}{t}\right) - 2\right)\]

  7. Simplified0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \color{blue}{\frac{1}{t}} \cdot \frac{\sqrt[3]{1}}{z}, \frac{2}{t}\right) - 2\right)\]

  8. Simplified0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{1}{t} \cdot \color{blue}{\frac{1}{z}}, \frac{2}{t}\right) - 2\right)\]
  9. Using strategy rm

  10. Applied associate-*l/0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \color{blue}{\frac{1 \cdot \frac{1}{z}}{t}}, \frac{2}{t}\right) - 2\right)\]

  11. Simplified0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\frac{1}{z}}}{t}, \frac{2}{t}\right) - 2\right)\]

  12. Final simplification0.1

    \[\leadsto \frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{z}}{t}, \frac{2}{t}\right) - 2\right)\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y)))

  (+ (/ x y) (/ (+ 2 (* (* z 2) (- 1 t))) (* t z))))