Average Error: 9.8 → 0.1
Time: 5.0s
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 r834669 = x;
        double r834670 = y;
        double r834671 = r834669 / r834670;
        double r834672 = 2.0;
        double r834673 = z;
        double r834674 = r834673 * r834672;
        double r834675 = 1.0;
        double r834676 = t;
        double r834677 = r834675 - r834676;
        double r834678 = r834674 * r834677;
        double r834679 = r834672 + r834678;
        double r834680 = r834676 * r834673;
        double r834681 = r834679 / r834680;
        double r834682 = r834671 + r834681;
        return r834682;
}
↓
double f(double x, double y, double z, double t) {
        double r834683 = 1.0;
        double r834684 = z;
        double r834685 = r834683 / r834684;
        double r834686 = 1.0;
        double r834687 = r834685 + r834686;
        double r834688 = t;
        double r834689 = r834687 - r834688;
        double r834690 = 2.0;
        double r834691 = r834690 / r834688;
        double r834692 = x;
        double r834693 = y;
        double r834694 = r834692 / r834693;
        double r834695 = fma(r834689, r834691, r834694);
        return r834695;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.8
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.8

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

  2. Simplified9.7

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

  3. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))))