Average Error: 9.3 → 0.1
Time: 3.5s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r812098 = x;
        double r812099 = y;
        double r812100 = r812098 / r812099;
        double r812101 = 2.0;
        double r812102 = z;
        double r812103 = r812102 * r812101;
        double r812104 = 1.0;
        double r812105 = t;
        double r812106 = r812104 - r812105;
        double r812107 = r812103 * r812106;
        double r812108 = r812101 + r812107;
        double r812109 = r812105 * r812102;
        double r812110 = r812108 / r812109;
        double r812111 = r812100 + r812110;
        return r812111;
}
↓
double f(double x, double y, double z, double t) {
        double r812112 = x;
        double r812113 = y;
        double r812114 = r812112 / r812113;
        double r812115 = 2.0;
        double r812116 = 1.0;
        double r812117 = t;
        double r812118 = z;
        double r812119 = r812117 * r812118;
        double r812120 = r812116 / r812119;
        double r812121 = r812116 / r812117;
        double r812122 = r812115 * r812121;
        double r812123 = r812122 - r812115;
        double r812124 = fma(r812115, r812120, r812123);
        double r812125 = r812114 + r812124;
        return r812125;
}

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. 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

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020024 +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))))