Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Average Error: 9.2 → 0.1

Time: 4.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r751116 = x;
        double r751117 = y;
        double r751118 = r751116 / r751117;
        double r751119 = 2.0;
        double r751120 = z;
        double r751121 = r751120 * r751119;
        double r751122 = 1.0;
        double r751123 = t;
        double r751124 = r751122 - r751123;
        double r751125 = r751121 * r751124;
        double r751126 = r751119 + r751125;
        double r751127 = r751123 * r751120;
        double r751128 = r751126 / r751127;
        double r751129 = r751118 + r751128;
        return r751129;
}

↓

double f(double x, double y, double z, double t) {
        double r751130 = x;
        double r751131 = y;
        double r751132 = r751130 / r751131;
        double r751133 = 2.0;
        double r751134 = 1.0;
        double r751135 = t;
        double r751136 = z;
        double r751137 = r751135 * r751136;
        double r751138 = r751134 / r751137;
        double r751139 = r751134 / r751135;
        double r751140 = r751133 * r751139;
        double r751141 = r751140 - r751133;
        double r751142 = fma(r751133, r751138, r751141);
        double r751143 = r751132 + r751142;
        return r751143;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.2

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

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. Simplified 0.1

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

4. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020001 +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))))