Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.7 → 10.7

Time: 17.5min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.13818177103117769 \cdot 10^{136}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((z <= -2.1381817710311777e+136)) {
		VAR = ((double) (((double) ((y / z) * ((double) (x - t)))) + t));
	} else {
		VAR = ((double) (x + (((double) (t - x)) / (((double) (a - z)) / ((double) (y - z))))));
	}
	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	24.7
Target	11.9
Herbie	10.7

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -2.13818177103117769e136

   1. Initial program 47.5

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

   2. Taylor expanded around inf 24.2

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

   3. Simplified 15.6

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

   if -2.13818177103117769e136 < z

   1. Initial program 20.6

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

   3. Applied add-cube-cbrt 21.1

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

   4. Applied times-frac 10.7

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied pow1 10.7

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

   7. Applied pow1 10.7

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

   8. Applied pow-prod-down 10.7

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

   9. Simplified 9.8

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.13818177103117769 \cdot 10^{136}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x - t\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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