Average Error: 9.3 → 0.1
Time: 11.6s
Precision: 64
\[\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}\]
↓
\[\mathsf{fma}\left(\frac{2.0}{t}, \left(1.0 + \frac{1}{z}\right) - t, \frac{x}{y}\right)\]
\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(\frac{2.0}{t}, \left(1.0 + \frac{1}{z}\right) - t, \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r15371039 = x;
        double r15371040 = y;
        double r15371041 = r15371039 / r15371040;
        double r15371042 = 2.0;
        double r15371043 = z;
        double r15371044 = r15371043 * r15371042;
        double r15371045 = 1.0;
        double r15371046 = t;
        double r15371047 = r15371045 - r15371046;
        double r15371048 = r15371044 * r15371047;
        double r15371049 = r15371042 + r15371048;
        double r15371050 = r15371046 * r15371043;
        double r15371051 = r15371049 / r15371050;
        double r15371052 = r15371041 + r15371051;
        return r15371052;
}
↓
double f(double x, double y, double z, double t) {
        double r15371053 = 2.0;
        double r15371054 = t;
        double r15371055 = r15371053 / r15371054;
        double r15371056 = 1.0;
        double r15371057 = 1.0;
        double r15371058 = z;
        double r15371059 = r15371057 / r15371058;
        double r15371060 = r15371056 + r15371059;
        double r15371061 = r15371060 - r15371054;
        double r15371062 = x;
        double r15371063 = y;
        double r15371064 = r15371062 / r15371063;
        double r15371065 = fma(r15371055, r15371061, r15371064);
        return r15371065;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.1
Herbie0.1
\[\frac{\frac{2.0}{z} + 2.0}{t} - \left(2.0 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.3

    \[\frac{x}{y} + \frac{2.0 + \left(z \cdot 2.0\right) \cdot \left(1.0 - t\right)}{t \cdot z}\]

  2. Simplified9.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2.0}{t}, \frac{\mathsf{fma}\left(z, 1.0 - t, 1\right)}{z}, \frac{x}{y}\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \mathsf{fma}\left(\frac{2.0}{t}, \color{blue}{\left(1.0 + \frac{1}{z}\right) - t}, \frac{x}{y}\right)\]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\frac{2.0}{t}, \left(1.0 + \frac{1}{z}\right) - t, \frac{x}{y}\right)\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))