Average Error: 9.6 → 0.1
Time: 19.9s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{\mathsf{fma}\left(2, 1, \frac{2}{z}\right)}{t} + \left(\frac{x}{y} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{\mathsf{fma}\left(2, 1, \frac{2}{z}\right)}{t} + \left(\frac{x}{y} - 2\right)
double f(double x, double y, double z, double t) {
        double r608111 = x;
        double r608112 = y;
        double r608113 = r608111 / r608112;
        double r608114 = 2.0;
        double r608115 = z;
        double r608116 = r608115 * r608114;
        double r608117 = 1.0;
        double r608118 = t;
        double r608119 = r608117 - r608118;
        double r608120 = r608116 * r608119;
        double r608121 = r608114 + r608120;
        double r608122 = r608118 * r608115;
        double r608123 = r608121 / r608122;
        double r608124 = r608113 + r608123;
        return r608124;
}
↓
double f(double x, double y, double z, double t) {
        double r608125 = 2.0;
        double r608126 = 1.0;
        double r608127 = z;
        double r608128 = r608125 / r608127;
        double r608129 = fma(r608125, r608126, r608128);
        double r608130 = t;
        double r608131 = r608129 / r608130;
        double r608132 = x;
        double r608133 = y;
        double r608134 = r608132 / r608133;
        double r608135 = r608134 - r608125;
        double r608136 = r608131 + r608135;
        return r608136;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +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))))