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

Average Error: 24.9 → 10.6

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -4.50268869983393748 \cdot 10^{-105} \lor \neg \left(a \le 1.2502224442482174 \cdot 10^{-160}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r553140 = x;
        double r553141 = y;
        double r553142 = r553141 - r553140;
        double r553143 = z;
        double r553144 = t;
        double r553145 = r553143 - r553144;
        double r553146 = r553142 * r553145;
        double r553147 = a;
        double r553148 = r553147 - r553144;
        double r553149 = r553146 / r553148;
        double r553150 = r553140 + r553149;
        return r553150;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r553151 = a;
        double r553152 = -4.5026886998339375e-105;
        bool r553153 = r553151 <= r553152;
        double r553154 = 1.2502224442482174e-160;
        bool r553155 = r553151 <= r553154;
        double r553156 = !r553155;
        bool r553157 = r553153 || r553156;
        double r553158 = x;
        double r553159 = y;
        double r553160 = r553159 - r553158;
        double r553161 = t;
        double r553162 = r553151 - r553161;
        double r553163 = z;
        double r553164 = r553163 - r553161;
        double r553165 = r553162 / r553164;
        double r553166 = r553160 / r553165;
        double r553167 = r553158 + r553166;
        double r553168 = r553158 * r553163;
        double r553169 = r553168 / r553161;
        double r553170 = r553159 + r553169;
        double r553171 = r553163 * r553159;
        double r553172 = r553171 / r553161;
        double r553173 = r553170 - r553172;
        double r553174 = r553157 ? r553167 : r553173;
        return r553174;
}

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	9.5
Herbie	10.6

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

Derivation

1. Split input into 2 regimes

2. if a < -4.5026886998339375e-105 or 1.2502224442482174e-160 < a

   1. Initial program 23.2

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

   3. Applied associate-/l* 9.0

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

   if -4.5026886998339375e-105 < a < 1.2502224442482174e-160

   1. Initial program 30.0

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

   2. Taylor expanded around inf 15.2

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.50268869983393748 \cdot 10^{-105} \lor \neg \left(a \le 1.2502224442482174 \cdot 10^{-160}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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))))