Average Error: 11.1 → 1.3
Time: 3.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{z - t}{z - a} \cdot y + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{z - t}{z - a} \cdot y + x
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 ((((z - t) / (z - a)) * y) + 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

Original11.1
Target1.1
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 11.1

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

  2. Simplified2.9

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

  4. Applied clear-num3.2

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{z - a}{y}}}, z - t, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef3.2

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

  7. Simplified3.0

    \[\leadsto \color{blue}{\frac{z - t}{\frac{z - a}{y}}} + x\]
  8. Using strategy rm

  9. Applied associate-/r/1.3

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

  10. Final simplification1.3

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))