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

Average Error: 9.2 → 0.1

Time: 4.2s

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 r736503 = x;
        double r736504 = y;
        double r736505 = r736503 / r736504;
        double r736506 = 2.0;
        double r736507 = z;
        double r736508 = r736507 * r736506;
        double r736509 = 1.0;
        double r736510 = t;
        double r736511 = r736509 - r736510;
        double r736512 = r736508 * r736511;
        double r736513 = r736506 + r736512;
        double r736514 = r736510 * r736507;
        double r736515 = r736513 / r736514;
        double r736516 = r736505 + r736515;
        return r736516;
}

↓

double f(double x, double y, double z, double t) {
        double r736517 = x;
        double r736518 = y;
        double r736519 = r736517 / r736518;
        double r736520 = 2.0;
        double r736521 = 1.0;
        double r736522 = t;
        double r736523 = z;
        double r736524 = r736522 * r736523;
        double r736525 = r736521 / r736524;
        double r736526 = r736521 / r736522;
        double r736527 = r736520 * r736526;
        double r736528 = r736527 - r736520;
        double r736529 = fma(r736520, r736525, r736528);
        double r736530 = r736519 + r736529;
        return r736530;
}

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))))