Average Error: 8.6 → 0.3
Time: 8.0s
Precision: binary64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
↓
\[\mathsf{fma}\left(x, \frac{1}{y}, -2 - \frac{-2 + \frac{-2}{z}}{t}\right) \]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\mathsf{fma}\left(x, \frac{1}{y}, -2 - \frac{-2 + \frac{-2}{z}}{t}\right)

(FPCore (x y z t)
 :precision binary64
 (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (fma x (/ 1.0 y) (- -2.0 (/ (+ -2.0 (/ -2.0 z)) t))))
double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}
↓
double code(double x, double y, double z, double t) {
	return fma(x, (1.0 / y), (-2.0 - ((-2.0 + (-2.0 / z)) / t)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original8.6
Target0.5
Herbie0.3
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right) \]

Derivation

  1. Initial program 8.6

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

  2. Simplified0.5

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

  3. Applied div-inv_binary640.6

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

  4. Applied fma-def_binary640.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2.0 z) 2.0) t) (- 2.0 (/ x y)))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))