Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A

Average Error: 10.9 → 1.3

Time: 2.4s

Precision: 64

Math

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

↓

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

C

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 fma(((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

Target

Original	10.9
Target	1.3
Herbie	1.3

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

Derivation

1. Initial program 10.9

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

2. Simplified 3.0

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

4. Applied clear-num 3.2

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

6. Applied fma-udef 3.2

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

7. Simplified 3.0

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

9. Applied frac-2neg 3.0

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

10. Applied associate-/r/ 1.3

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

11. Applied fma-def 1.3

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

12. Final simplification 1.3

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

Reproduce

herbie shell --seed 2020071 +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))))