Average Error: 9.4 → 0.2
Time: 5.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\left(\frac{1}{z} + 1\right) - t, \frac{2}{t}, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r1079785 = x;
        double r1079786 = y;
        double r1079787 = r1079785 / r1079786;
        double r1079788 = 2.0;
        double r1079789 = z;
        double r1079790 = r1079789 * r1079788;
        double r1079791 = 1.0;
        double r1079792 = t;
        double r1079793 = r1079791 - r1079792;
        double r1079794 = r1079790 * r1079793;
        double r1079795 = r1079788 + r1079794;
        double r1079796 = r1079792 * r1079789;
        double r1079797 = r1079795 / r1079796;
        double r1079798 = r1079787 + r1079797;
        return r1079798;
}
↓
double f(double x, double y, double z, double t) {
        double r1079799 = 1.0;
        double r1079800 = z;
        double r1079801 = r1079799 / r1079800;
        double r1079802 = 1.0;
        double r1079803 = r1079801 + r1079802;
        double r1079804 = t;
        double r1079805 = r1079803 - r1079804;
        double r1079806 = 2.0;
        double r1079807 = r1079806 / r1079804;
        double r1079808 = x;
        double r1079809 = y;
        double r1079810 = r1079808 / r1079809;
        double r1079811 = fma(r1079805, r1079807, r1079810);
        return r1079811;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.4
Target0.1
Herbie0.2
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.4

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

  2. Simplified9.4

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

  3. Taylor expanded around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +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))))