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

Average Error: 10.9 → 1.5

Time: 6.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0071455594193599 \cdot 10^{-271} \lor \neg \left(t \le 8.49657610056407139 \cdot 10^{-71}\right):\\ \;\;\;\;y \cdot \left(\frac{z}{a - t} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t}{a - t}\right)\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t} + x\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return (x + ((y * (z - t)) / (a - t)));
}

↓

double code(double x, double y, double z, double t, double a) {
	double temp;
	if (((t <= -1.0071455594193599e-271) || !(t <= 8.496576100564071e-71))) {
		temp = ((y * ((z / (a - t)) - log1p(expm1((t / (a - t)))))) + x);
	} else {
		temp = (((y * (z - t)) / (a - t)) + x);
	}
	return temp;
}

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	1.5

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

Derivation

1. Split input into 2 regimes

2. if t < -1.0071455594193599e-271 or 8.496576100564071e-71 < t

   1. Initial program 12.7

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

   2. Simplified 2.9

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

   4. Applied fma-udef 2.9

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

   6. Applied div-inv 2.9

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

   7. Applied associate-*l* 0.9

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

   8. Simplified 0.9

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

   10. Applied div-sub 0.8

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

   12. Applied log1p-expm1-u 0.9

       \[\leadsto y \cdot \left(\frac{z}{a - t} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t}{a - t}\right)\right)}\right) + x\]

   if -1.0071455594193599e-271 < t < 8.496576100564071e-71

   1. Initial program 3.8

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

   2. Simplified 3.5

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

   4. Applied fma-udef 3.5

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

   6. Applied div-inv 3.5

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

   7. Applied associate-*l* 3.5

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

   8. Simplified 3.5

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

   10. Applied associate-*r/ 3.8

       \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} + x\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0071455594193599 \cdot 10^{-271} \lor \neg \left(t \le 8.49657610056407139 \cdot 10^{-71}\right):\\ \;\;\;\;y \cdot \left(\frac{z}{a - t} - \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t}{a - t}\right)\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a - t} + x\\ \end{array}\]

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, B"
  :precision binary64

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

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