Average Error: 6.0 → 2.4
Time: 4.7s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a} \]
↓
\[\mathsf{fma}\left(\frac{y}{a}, t - z, x\right) \]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)

(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a) :precision binary64 (fma (/ y a) (- t z) x))
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

Target

Original6.0
Target0.7
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

  1. Initial program 6.0

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

  2. Simplified6.1

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

  3. Taylor expanded in y around 0 6.0

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

  4. Simplified2.4

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

  5. Final simplification2.4

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

Reproduce

herbie shell --seed 2021280 
(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 (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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