Average Error: 5.9 → 2.7
Time: 1.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)
double code(double x, double y, double z, double t, double a) {
	return (x - ((y * (z - t)) / a));
}
↓
double code(double x, double y, double z, double t, double a) {
	return fma((y / a), (t - z), x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.9
Target0.7
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 5.9

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

  2. Simplified2.7

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

  3. Final simplification2.7

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))