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

Average Error: 10.6 → 0.7

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \le -25.915352012834369:\\ \;\;\;\;\frac{t}{-\left(a - z\right)} \cdot \left(-\left(y - z\right)\right) + x\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \le 1.3713989973819208 \cdot 10^{285}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (x + ((double) (((double) (((double) (y - z)) * t)) / ((double) (a - z)))))) <= -25.91535201283437)) {
		VAR = ((double) (((double) (((double) (t / ((double) -(((double) (a - z)))))) * ((double) -(((double) (y - z)))))) + x));
	} else {
		double VAR_1;
		if ((((double) (x + ((double) (((double) (((double) (y - z)) * t)) / ((double) (a - z)))))) <= 1.3713989973819208e+285)) {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y - z)) * t)) / ((double) (a - z))))));
		} else {
			VAR_1 = ((double) fma(((double) (((double) (y - z)) / ((double) (a - z)))), t, x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	0.5
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y z) t) (- a z))) < -25.91535201283437

   1. Initial program 14.8

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

   2. Simplified 1.8

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

   4. Applied div-inv 1.9

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

   6. Applied add-cube-cbrt 2.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}\right)} \cdot \frac{1}{a - z}, t, x\right)\]

   7. Applied associate-*l* 2.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{1}{a - z}\right)}, t, x\right)\]

   8. Simplified 2.3

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

   10. Applied fma-udef 2.3

       \[\leadsto \color{blue}{\left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \frac{\sqrt[3]{y - z}}{a - z}\right) \cdot t + x}\]

   11. Simplified 1.6

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

   13. Applied frac-2neg 1.6

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

   14. Applied associate-/r/ 1.3

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

   if -25.91535201283437 < (+ x (/ (* (- y z) t) (- a z))) < 1.3713989973819208e+285

   1. Initial program 0.3

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

   if 1.3713989973819208e+285 < (+ x (/ (* (- y z) t) (- a z)))

   1. Initial program 51.9

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

   2. Simplified 0.8

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \le -25.915352012834369:\\ \;\;\;\;\frac{t}{-\left(a - z\right)} \cdot \left(-\left(y - z\right)\right) + x\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot t}{a - z} \le 1.3713989973819208 \cdot 10^{285}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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