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

Average Error: 24.3 → 9.7

Time: 25.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a - t}, z - t, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.00144295601552635 \cdot 10^{-245}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 9.1139 \cdot 10^{-305}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 4.67784614199411992 \cdot 10^{290}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a - t}, z - t, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r378745 = x;
        double r378746 = y;
        double r378747 = r378746 - r378745;
        double r378748 = z;
        double r378749 = t;
        double r378750 = r378748 - r378749;
        double r378751 = r378747 * r378750;
        double r378752 = a;
        double r378753 = r378752 - r378749;
        double r378754 = r378751 / r378753;
        double r378755 = r378745 + r378754;
        return r378755;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r378756 = x;
        double r378757 = y;
        double r378758 = r378757 - r378756;
        double r378759 = z;
        double r378760 = t;
        double r378761 = r378759 - r378760;
        double r378762 = r378758 * r378761;
        double r378763 = a;
        double r378764 = r378763 - r378760;
        double r378765 = r378762 / r378764;
        double r378766 = r378756 + r378765;
        double r378767 = -inf.0;
        bool r378768 = r378766 <= r378767;
        double r378769 = r378758 / r378764;
        double r378770 = fma(r378769, r378761, r378756);
        double r378771 = -3.0014429560155264e-245;
        bool r378772 = r378766 <= r378771;
        double r378773 = 9.113902524445497e-305;
        bool r378774 = r378766 <= r378773;
        double r378775 = 4.67784614199412e+290;
        bool r378776 = r378766 <= r378775;
        double r378777 = r378776 ? r378766 : r378770;
        double r378778 = r378774 ? r378757 : r378777;
        double r378779 = r378772 ? r378766 : r378778;
        double r378780 = r378768 ? r378770 : r378779;
        return r378780;
}

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	9.5
Herbie	9.7

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -inf.0 or 4.67784614199412e+290 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 62.1

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

   2. Simplified 17.6

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

   if -inf.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -3.0014429560155264e-245 or 9.113902524445497e-305 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 4.67784614199412e+290

   1. Initial program 2.0

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

   if -3.0014429560155264e-245 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 9.113902524445497e-305

   1. Initial program 54.3

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

   2. Simplified 55.3

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

   3. Taylor expanded around 0 36.4

      \[\leadsto \color{blue}{y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a - t}, z - t, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.00144295601552635 \cdot 10^{-245}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 9.1139 \cdot 10^{-305}:\\ \;\;\;\;y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 4.67784614199411992 \cdot 10^{290}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a - t}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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