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

Average Error: 24.5 → 11.9

Time: 4.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.36070110252963308 \cdot 10^{191} \lor \neg \left(t \le 2.05294572640722941 \cdot 10^{202}\right):\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y - x}}} \cdot \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{a - t}}{z - t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((t <= -5.360701102529633e+191) || !(t <= 2.0529457264072294e+202))) {
		VAR = ((double) (((double) (y + ((double) (((double) (x * z)) / t)))) - ((double) (((double) (z * y)) / t))));
	} else {
		VAR = ((double) (x + ((double) (((double) (((double) cbrt(((double) (y - x)))) / ((double) (((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))) / ((double) cbrt(((double) (y - x)))))))) * ((double) (((double) cbrt(((double) (y - x)))) / ((double) (((double) cbrt(((double) (a - t)))) / ((double) (z - 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	24.5
Target	9.5
Herbie	11.9

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -5.36070110252963308e191 or 2.05294572640722941e202 < t

   1. Initial program 49.9

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

   2. Taylor expanded around inf 24.0

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

   if -5.36070110252963308e191 < t < 2.05294572640722941e202

   1. Initial program 17.8

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

   3. Applied associate-/l* 8.3

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

   5. Applied *-un-lft-identity 8.3

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

   6. Applied add-cube-cbrt 8.9

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

   7. Applied times-frac 8.9

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

   8. Applied add-cube-cbrt 9.0

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

   9. Applied times-frac 8.7

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

   10. Simplified 8.7

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

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.36070110252963308 \cdot 10^{191} \lor \neg \left(t \le 2.05294572640722941 \cdot 10^{202}\right):\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y - x}}} \cdot \frac{\sqrt[3]{y - x}}{\frac{\sqrt[3]{a - t}}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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