Average Error: 8.9 → 0.1
Time: 25.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(2, 1, \frac{2}{z}\right), \frac{1}{t}, \frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\mathsf{fma}\left(2, 1, \frac{2}{z}\right), \frac{1}{t}, \frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r492585 = x;
        double r492586 = y;
        double r492587 = r492585 / r492586;
        double r492588 = 2.0;
        double r492589 = z;
        double r492590 = r492589 * r492588;
        double r492591 = 1.0;
        double r492592 = t;
        double r492593 = r492591 - r492592;
        double r492594 = r492590 * r492593;
        double r492595 = r492588 + r492594;
        double r492596 = r492592 * r492589;
        double r492597 = r492595 / r492596;
        double r492598 = r492587 + r492597;
        return r492598;
}
↓
double f(double x, double y, double z, double t) {
        double r492599 = 2.0;
        double r492600 = 1.0;
        double r492601 = z;
        double r492602 = r492599 / r492601;
        double r492603 = fma(r492599, r492600, r492602);
        double r492604 = 1.0;
        double r492605 = t;
        double r492606 = r492604 / r492605;
        double r492607 = x;
        double r492608 = y;
        double r492609 = r492607 / r492608;
        double r492610 = r492609 - r492599;
        double r492611 = fma(r492603, r492606, r492610);
        return r492611;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original8.9
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 8.9

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

  2. Simplified0.1

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

  4. Applied div-inv0.1

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

  5. Applied fma-def0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019347 +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))))