Average Error: 10.6 → 1.1
Time: 5.7s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[x + \frac{y}{\frac{z - a}{z - t}} \]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
x + \frac{y}{\frac{z - a}{z - t}}

(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))
↓
(FPCore (x y z t a) :precision binary64 (+ x (/ y (/ (- z a) (- z t)))))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}
↓
double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - t)));
}

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

Original10.6
Target1.1
Herbie1.1
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Initial program 10.6

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

  2. Simplified1.3

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

  3. Taylor expanded in y around 0 10.6

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

  4. Simplified2.8

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

  5. Applied egg-rr1.1

    \[\leadsto x + \color{blue}{\frac{y}{\frac{z - a}{z - t}}} \]

  6. Final simplification1.1

    \[\leadsto x + \frac{y}{\frac{z - a}{z - t}} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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