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

Average Error: 24.3 → 8.2

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.36370666452502506 \cdot 10^{-267} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((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 (((((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))))) <= -1.363706664525025e-267) || !(((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))))) <= 0.0))) {
		VAR = ((double) (x + ((double) (((double) (t - x)) * ((double) (((double) (y - z)) / ((double) (a - z))))))));
	} else {
		VAR = ((double) (t + ((double) (((double) (y / z)) * ((double) (x - t))))));
	}
	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	24.3
Target	12.0
Herbie	8.2

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -1.36370666452502506e-267 or 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 21.2

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

   2. Simplified 10.5

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

   4. Applied add-cube-cbrt 11.2

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

   5. Applied *-un-lft-identity 11.2

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

   6. Applied times-frac 11.2

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

   7. Applied associate-*r* 8.2

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

   8. Simplified 8.2

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

   10. Applied pow1 8.2

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

   11. Applied pow1 8.2

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

   12. Applied pow-prod-down 8.2

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

   13. Simplified 7.1

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

   if -1.36370666452502506e-267 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program 58.1

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

   2. Simplified 58.2

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

   3. Taylor expanded around inf 20.0

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

   4. Simplified 20.0

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

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.36370666452502506 \cdot 10^{-267} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \end{array}\]

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))))