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

Average Error: 24.9 → 10.6

Time: 21.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 9.405156079127402373396803314253208936876 \cdot 10^{225}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} - \frac{t}{z}\right) \cdot y + t\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r462242 = x;
        double r462243 = y;
        double r462244 = z;
        double r462245 = r462243 - r462244;
        double r462246 = t;
        double r462247 = r462246 - r462242;
        double r462248 = r462245 * r462247;
        double r462249 = a;
        double r462250 = r462249 - r462244;
        double r462251 = r462248 / r462250;
        double r462252 = r462242 + r462251;
        return r462252;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r462253 = z;
        double r462254 = 9.405156079127402e+225;
        bool r462255 = r462253 <= r462254;
        double r462256 = x;
        double r462257 = t;
        double r462258 = r462257 - r462256;
        double r462259 = a;
        double r462260 = r462259 - r462253;
        double r462261 = y;
        double r462262 = r462261 - r462253;
        double r462263 = r462260 / r462262;
        double r462264 = r462258 / r462263;
        double r462265 = r462256 + r462264;
        double r462266 = r462256 / r462253;
        double r462267 = r462257 / r462253;
        double r462268 = r462266 - r462267;
        double r462269 = r462268 * r462261;
        double r462270 = r462269 + r462257;
        double r462271 = r462255 ? r462265 : r462270;
        return r462271;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.9
Target	11.8
Herbie	10.6

\[\begin{array}{l} \mathbf{if}\;z \lt -1.253613105609503593846459977496550767343 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < 9.405156079127402e+225

   1. Initial program 22.4

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

   2. Simplified 10.3

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

   4. Applied fma-udef 10.3

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

   5. Simplified 10.3

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

   if 9.405156079127402e+225 < z

   1. Initial program 50.7

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

   2. Simplified 25.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)}\]

   3. Taylor expanded around inf 21.6

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

   4. Simplified 13.6

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 9.405156079127402373396803314253208936876 \cdot 10^{225}:\\ \;\;\;\;x + \frac{t - x}{\frac{a - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} - \frac{t}{z}\right) \cdot y + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :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))))