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

Average Error: 24.3 → 8.5

Time: 6.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.29505595240201798 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r612359 = x;
        double r612360 = y;
        double r612361 = r612360 - r612359;
        double r612362 = z;
        double r612363 = t;
        double r612364 = r612362 - r612363;
        double r612365 = r612361 * r612364;
        double r612366 = a;
        double r612367 = r612366 - r612363;
        double r612368 = r612365 / r612367;
        double r612369 = r612359 + r612368;
        return r612369;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r612370 = x;
        double r612371 = y;
        double r612372 = r612371 - r612370;
        double r612373 = z;
        double r612374 = t;
        double r612375 = r612373 - r612374;
        double r612376 = r612372 * r612375;
        double r612377 = a;
        double r612378 = r612377 - r612374;
        double r612379 = r612376 / r612378;
        double r612380 = r612370 + r612379;
        double r612381 = -3.295055952402018e-290;
        bool r612382 = r612380 <= r612381;
        double r612383 = 0.0;
        bool r612384 = r612380 <= r612383;
        double r612385 = !r612384;
        bool r612386 = r612382 || r612385;
        double r612387 = r612375 / r612378;
        double r612388 = r612372 * r612387;
        double r612389 = r612370 + r612388;
        double r612390 = r612370 * r612373;
        double r612391 = r612390 / r612374;
        double r612392 = r612371 + r612391;
        double r612393 = r612373 * r612371;
        double r612394 = r612393 / r612374;
        double r612395 = r612392 - r612394;
        double r612396 = r612386 ? r612389 : r612395;
        return r612396;
}

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	9.4
Herbie	8.5

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -3.295055952402018e-290 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.2

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

   3. Applied *-un-lft-identity 21.2

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

   4. Applied times-frac 7.5

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

   5. Simplified 7.5

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

   if -3.295055952402018e-290 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 59.7

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

   2. Taylor expanded around inf 19.3

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.29505595240201798 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))