Average Error: 9.4 → 0.3
Time: 3.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \left(\sqrt[3]{\frac{1}{t \cdot z}} \cdot \sqrt[3]{\frac{1}{t \cdot z}}\right) \cdot \sqrt[3]{\frac{1}{t \cdot z}}, 2 \cdot \frac{1}{t} - 2\right)\]
\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, \left(\sqrt[3]{\frac{1}{t \cdot z}} \cdot \sqrt[3]{\frac{1}{t \cdot z}}\right) \cdot \sqrt[3]{\frac{1}{t \cdot z}}, 2 \cdot \frac{1}{t} - 2\right)
double code(double x, double y, double z, double t) {
	return ((double) (((double) (x / y)) + ((double) (((double) (2.0 + ((double) (((double) (z * 2.0)) * ((double) (1.0 - t)))))) / ((double) (t * z))))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (x / y)) + ((double) fma(2.0, ((double) (((double) (((double) cbrt(((double) (1.0 / ((double) (t * z)))))) * ((double) cbrt(((double) (1.0 / ((double) (t * z)))))))) * ((double) cbrt(((double) (1.0 / ((double) (t * z)))))))), ((double) (((double) (2.0 * ((double) (1.0 / t)))) - 2.0))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.1
Herbie0.3
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.4

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

  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. Using strategy rm

  5. Applied add-cube-cbrt0.3

    \[\leadsto \frac{x}{y} + \mathsf{fma}\left(2, \color{blue}{\left(\sqrt[3]{\frac{1}{t \cdot z}} \cdot \sqrt[3]{\frac{1}{t \cdot z}}\right) \cdot \sqrt[3]{\frac{1}{t \cdot z}}}, 2 \cdot \frac{1}{t} - 2\right)\]

  6. Final simplification0.3

    \[\leadsto \frac{x}{y} + \mathsf{fma}\left(2, \left(\sqrt[3]{\frac{1}{t \cdot z}} \cdot \sqrt[3]{\frac{1}{t \cdot z}}\right) \cdot \sqrt[3]{\frac{1}{t \cdot z}}, 2 \cdot \frac{1}{t} - 2\right)\]

Reproduce

herbie shell --seed 2020121 +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))))