Average Error: 9.3 → 0.1
Time: 4.9s
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 r870104 = x;
        double r870105 = y;
        double r870106 = r870104 / r870105;
        double r870107 = 2.0;
        double r870108 = z;
        double r870109 = r870108 * r870107;
        double r870110 = 1.0;
        double r870111 = t;
        double r870112 = r870110 - r870111;
        double r870113 = r870109 * r870112;
        double r870114 = r870107 + r870113;
        double r870115 = r870111 * r870108;
        double r870116 = r870114 / r870115;
        double r870117 = r870106 + r870116;
        return r870117;
}
↓
double f(double x, double y, double z, double t) {
        double r870118 = 1.0;
        double r870119 = z;
        double r870120 = r870118 / r870119;
        double r870121 = 1.0;
        double r870122 = r870120 + r870121;
        double r870123 = t;
        double r870124 = r870122 - r870123;
        double r870125 = 2.0;
        double r870126 = r870125 / r870123;
        double r870127 = x;
        double r870128 = y;
        double r870129 = r870127 / r870128;
        double r870130 = fma(r870124, r870126, r870129);
        return r870130;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.3
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.3

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

  2. Simplified9.3

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

  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 2020062 +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))))