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

Average Error: 1.5 → 1.5

Time: 12.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.3956560300111874 \cdot 10^{105}:\\ \;\;\;\;x + \frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a} - \frac{t}{z - a}, y, x\right) + y \cdot \mathsf{fma}\left(-\frac{t}{z - a}, 1, \frac{t}{z - a} \cdot 1\right)\\ \end{array}\]

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) {
	double temp;
	if ((x <= -9.395656030011187e+105)) {
		temp = (x + ((y / (z - a)) * (z - t)));
	} else {
		temp = (fma(((z / (z - a)) - (t / (z - a))), y, x) + (y * fma(-(t / (z - a)), 1.0, ((t / (z - a)) * 1.0))));
	}
	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	1.5
Target	1.4
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if x < -9.395656030011187e+105

   1. Initial program 1.1

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

   3. Applied div-sub 1.1

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

   5. Applied div-inv 1.1

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

   6. Applied div-inv 1.1

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

   7. Applied distribute-rgt-out-- 1.1

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

   8. Applied associate-*r* 0.8

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

   9. Simplified 0.8

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

   if -9.395656030011187e+105 < x

   1. Initial program 1.6

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

   3. Applied div-sub 1.6

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

   5. Applied *-un-lft-identity 1.6

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

   6. Applied *-un-lft-identity 1.6

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

   7. Applied times-frac 1.6

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

   8. Applied add-sqr-sqrt 13.6

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

   9. Applied prod-diff 13.6

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

   10. Applied distribute-lft-in 13.6

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

   11. Applied associate-+r+ 13.6

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

   12. Simplified 1.6

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{z - a} - \frac{t}{z - a}, y, x\right)} + y \cdot \mathsf{fma}\left(-\frac{t}{z - a}, \frac{1}{1}, \frac{t}{z - a} \cdot \frac{1}{1}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.3956560300111874 \cdot 10^{105}:\\ \;\;\;\;x + \frac{y}{z - a} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a} - \frac{t}{z - a}, y, x\right) + y \cdot \mathsf{fma}\left(-\frac{t}{z - a}, 1, \frac{t}{z - a} \cdot 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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