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

Average Error: 9.6 → 0.1

Time: 19.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r492111 = x;
        double r492112 = y;
        double r492113 = r492111 / r492112;
        double r492114 = 2.0;
        double r492115 = z;
        double r492116 = r492115 * r492114;
        double r492117 = 1.0;
        double r492118 = t;
        double r492119 = r492117 - r492118;
        double r492120 = r492116 * r492119;
        double r492121 = r492114 + r492120;
        double r492122 = r492118 * r492115;
        double r492123 = r492121 / r492122;
        double r492124 = r492113 + r492123;
        return r492124;
}

↓

double f(double x, double y, double z, double t) {
        double r492125 = 2.0;
        double r492126 = 1.0;
        double r492127 = z;
        double r492128 = r492125 / r492127;
        double r492129 = fma(r492125, r492126, r492128);
        double r492130 = t;
        double r492131 = r492129 / r492130;
        double r492132 = x;
        double r492133 = y;
        double r492134 = r492132 / r492133;
        double r492135 = r492134 - r492125;
        double r492136 = r492131 + r492135;
        return r492136;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.6
Target	0.1
Herbie	0.1

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

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. Simplified 0.1

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

3. Final simplification 0.1

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

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))))