Average Error: 9.5 → 0.1
Time: 3.3s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r2301674 = x;
        double r2301675 = y;
        double r2301676 = r2301674 / r2301675;
        double r2301677 = 2.0;
        double r2301678 = z;
        double r2301679 = r2301678 * r2301677;
        double r2301680 = 1.0;
        double r2301681 = t;
        double r2301682 = r2301680 - r2301681;
        double r2301683 = r2301679 * r2301682;
        double r2301684 = r2301677 + r2301683;
        double r2301685 = r2301681 * r2301678;
        double r2301686 = r2301684 / r2301685;
        double r2301687 = r2301676 + r2301686;
        return r2301687;
}
↓
double f(double x, double y, double z, double t) {
        double r2301688 = x;
        double r2301689 = y;
        double r2301690 = r2301688 / r2301689;
        double r2301691 = 2.0;
        double r2301692 = 1.0;
        double r2301693 = t;
        double r2301694 = z;
        double r2301695 = r2301693 * r2301694;
        double r2301696 = r2301692 / r2301695;
        double r2301697 = r2301692 / r2301693;
        double r2301698 = r2301691 * r2301697;
        double r2301699 = r2301698 - r2301691;
        double r2301700 = fma(r2301691, r2301696, r2301699);
        double r2301701 = r2301690 + r2301700;
        return r2301701;
}

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

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020018 +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))))