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

Average Error: 24.6 → 11.1

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.02461295362367818 \cdot 10^{-245}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;a \le 1.21960258162229716 \cdot 10^{-82}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{a - t}{\sqrt[3]{z - t}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((a <= -1.0246129536236782e-245)) {
		VAR = (x + ((y - x) / ((a - t) / (z - t))));
	} else {
		double VAR_1;
		if ((a <= 1.2196025816222972e-82)) {
			VAR_1 = ((y + ((x * z) / t)) - ((z * y) / t));
		} else {
			VAR_1 = (x + ((y - x) * ((cbrt((z - t)) * cbrt((z - t))) / ((a - t) / cbrt((z - t))))));
		}
		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	24.6
Target	9.4
Herbie	11.1

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

Derivation

1. Split input into 3 regimes

2. if a < -1.0246129536236782e-245

   1. Initial program 23.9

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

   3. Applied associate-/l* 11.0

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

   if -1.0246129536236782e-245 < a < 1.2196025816222972e-82

   1. Initial program 28.8

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

   2. Taylor expanded around inf 14.8

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

   if 1.2196025816222972e-82 < a

   1. Initial program 23.0

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

   3. Applied *-un-lft-identity 23.0

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

   4. Applied times-frac 8.5

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

   5. Simplified 8.5

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

   7. Applied add-cube-cbrt 9.0

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

   8. Applied associate-/l* 9.0

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.02461295362367818 \cdot 10^{-245}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;a \le 1.21960258162229716 \cdot 10^{-82}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{a - t}{\sqrt[3]{z - t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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