Average Error: 0.0 → 0.1
Time: 3.2s
Precision: binary64
\[\frac{\left(x + y\right) - z}{t \cdot 2} \]
↓
\[0.5 \cdot \left(\frac{y}{t} + \frac{x}{t}\right) - 0.5 \cdot \frac{z}{t} \]
\frac{\left(x + y\right) - z}{t \cdot 2}
↓
0.5 \cdot \left(\frac{y}{t} + \frac{x}{t}\right) - 0.5 \cdot \frac{z}{t}

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))
↓
(FPCore (x y z t)
 :precision binary64
 (- (* 0.5 (+ (/ y t) (/ x t))) (* 0.5 (/ z t))))
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 * ((y / t) + (x / t))) - (0.5 * (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 in x around 0 0.1

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

  3. Taylor expanded in x around inf 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022077 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))