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

Average Error: 25.2 → 12.1

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2.34703852665204316 \cdot 10^{195}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r721977 = x;
        double r721978 = y;
        double r721979 = r721978 - r721977;
        double r721980 = z;
        double r721981 = t;
        double r721982 = r721980 - r721981;
        double r721983 = r721979 * r721982;
        double r721984 = a;
        double r721985 = r721984 - r721981;
        double r721986 = r721983 / r721985;
        double r721987 = r721977 + r721986;
        return r721987;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r721988 = t;
        double r721989 = 2.347038526652043e+195;
        bool r721990 = r721988 <= r721989;
        double r721991 = x;
        double r721992 = y;
        double r721993 = r721992 - r721991;
        double r721994 = z;
        double r721995 = r721994 - r721988;
        double r721996 = a;
        double r721997 = r721996 - r721988;
        double r721998 = r721995 / r721997;
        double r721999 = r721993 * r721998;
        double r722000 = r721991 + r721999;
        double r722001 = r721991 * r721994;
        double r722002 = r722001 / r721988;
        double r722003 = r721992 + r722002;
        double r722004 = r721994 * r721992;
        double r722005 = r722004 / r721988;
        double r722006 = r722003 - r722005;
        double r722007 = r721990 ? r722000 : r722006;
        return r722007;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.2
Target	9.4
Herbie	12.1

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

Derivation

1. Split input into 2 regimes

2. if t < 2.347038526652043e+195

   1. Initial program 22.0

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

   3. Applied *-un-lft-identity 22.0

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

   4. Applied times-frac 10.5

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

   5. Simplified 10.5

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

   if 2.347038526652043e+195 < t

   1. Initial program 51.6

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

   2. Taylor expanded around inf 25.4

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

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.34703852665204316 \cdot 10^{195}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))))