Average Error: 9.6 → 0.1
Time: 4.8s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \left(\mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, \frac{2}{t}\right) - 2\right)\]
\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}{t}}{z}, \frac{2}{t}\right) - 2\right)
double f(double x, double y, double z, double t) {
        double r698640 = x;
        double r698641 = y;
        double r698642 = r698640 / r698641;
        double r698643 = 2.0;
        double r698644 = z;
        double r698645 = r698644 * r698643;
        double r698646 = 1.0;
        double r698647 = t;
        double r698648 = r698646 - r698647;
        double r698649 = r698645 * r698648;
        double r698650 = r698643 + r698649;
        double r698651 = r698647 * r698644;
        double r698652 = r698650 / r698651;
        double r698653 = r698642 + r698652;
        return r698653;
}
↓
double f(double x, double y, double z, double t) {
        double r698654 = x;
        double r698655 = y;
        double r698656 = r698654 / r698655;
        double r698657 = 2.0;
        double r698658 = 1.0;
        double r698659 = t;
        double r698660 = r698658 / r698659;
        double r698661 = z;
        double r698662 = r698660 / r698661;
        double r698663 = r698657 / r698659;
        double r698664 = fma(r698657, r698662, r698663);
        double r698665 = r698664 - r698657;
        double r698666 = r698656 + r698665;
        return r698666;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.6
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.6

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

  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 associate-/r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020034 +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))))