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

Average Error: 10.6 → 0.3

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -inf.0:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 2.1327535190838117 \cdot 10^{286}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z - a}}{\frac{1}{z - t}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.6
Target	1.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (/ (* y (- z t)) (- z a)) < -inf.0

   1. Initial program 64.0

      \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 64.0

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

   if -inf.0 < (/ (* y (- z t)) (- z a)) < 2.1327535190838117e286

   1. Initial program 0.2

      \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
   2. Using strategy rm

   3. Applied sub-neg 0.2

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

   4. Applied distribute-lft-in 0.2

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

   if 2.1327535190838117e286 < (/ (* y (- z t)) (- z a))

   1. Initial program 60.5

      \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.6

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

   5. Applied div-inv 0.7

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

   6. Applied associate-/r* 1.2

      \[\leadsto x + \color{blue}{\frac{\frac{y}{z - a}}{\frac{1}{z - t}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -inf.0:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 2.1327535190838117 \cdot 10^{286}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z - a}}{\frac{1}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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))))