Average Error: 0.0 → 0.1
Time: 5.0s
Precision: 64
\[\frac{\left(x + y\right) - z}{t \cdot 2}\]
↓
\[0.5 \cdot \left(\mathsf{fma}\left(y, \frac{1}{t}, \frac{x}{t}\right) - \frac{z}{t}\right)\]
\frac{\left(x + y\right) - z}{t \cdot 2}
↓
0.5 \cdot \left(\mathsf{fma}\left(y, \frac{1}{t}, \frac{x}{t}\right) - \frac{z}{t}\right)
double code(double x, double y, double z, double t) {
	return (((x + y) - z) / (t * 2.0));
}
↓
double code(double x, double y, double z, double t) {
	return (0.5 * (fma(y, (1.0 / t), (x / t)) - (z / t)));
}

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

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(0.5 \cdot \frac{y}{t} + 0.5 \cdot \frac{x}{t}\right) - 0.5 \cdot \frac{z}{t}}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(\frac{y}{t} + \frac{x}{t}\right) - \frac{z}{t}\right)}\]
  4. Using strategy rm

  5. Applied div-inv0.1

    \[\leadsto 0.5 \cdot \left(\left(\color{blue}{y \cdot \frac{1}{t}} + \frac{x}{t}\right) - \frac{z}{t}\right)\]

  6. Applied fma-def0.1

    \[\leadsto 0.5 \cdot \left(\color{blue}{\mathsf{fma}\left(y, \frac{1}{t}, \frac{x}{t}\right)} - \frac{z}{t}\right)\]

  7. Final simplification0.1

    \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(y, \frac{1}{t}, \frac{x}{t}\right) - \frac{z}{t}\right)\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2)))