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

Average Error: 24.5 → 8.7

Time: 15.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.153952848057346 \cdot 10^{-251} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;x + \left(\frac{z}{a - t} \cdot \left(y - x\right) + \left(-\frac{t}{a - t}\right) \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r540789 = x;
        double r540790 = y;
        double r540791 = r540790 - r540789;
        double r540792 = z;
        double r540793 = t;
        double r540794 = r540792 - r540793;
        double r540795 = r540791 * r540794;
        double r540796 = a;
        double r540797 = r540796 - r540793;
        double r540798 = r540795 / r540797;
        double r540799 = r540789 + r540798;
        return r540799;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r540800 = x;
        double r540801 = y;
        double r540802 = r540801 - r540800;
        double r540803 = z;
        double r540804 = t;
        double r540805 = r540803 - r540804;
        double r540806 = r540802 * r540805;
        double r540807 = a;
        double r540808 = r540807 - r540804;
        double r540809 = r540806 / r540808;
        double r540810 = r540800 + r540809;
        double r540811 = -2.153952848057346e-251;
        bool r540812 = r540810 <= r540811;
        double r540813 = 0.0;
        bool r540814 = r540810 <= r540813;
        double r540815 = !r540814;
        bool r540816 = r540812 || r540815;
        double r540817 = r540803 / r540808;
        double r540818 = r540817 * r540802;
        double r540819 = r540804 / r540808;
        double r540820 = -r540819;
        double r540821 = r540820 * r540802;
        double r540822 = r540818 + r540821;
        double r540823 = r540800 + r540822;
        double r540824 = r540800 * r540803;
        double r540825 = r540824 / r540804;
        double r540826 = r540801 + r540825;
        double r540827 = r540803 * r540801;
        double r540828 = r540827 / r540804;
        double r540829 = r540826 - r540828;
        double r540830 = r540816 ? r540823 : r540829;
        return r540830;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	9.2
Herbie	8.7

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -2.153952848057346e-251 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.6

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

   3. Applied *-un-lft-identity 21.6

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

   4. Applied times-frac 7.5

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

   5. Simplified 7.5

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

   7. Applied div-sub 7.5

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

   9. Applied sub-neg 7.5

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

   10. Applied distribute-lft-in 7.5

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

   11. Simplified 7.5

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

   12. Simplified 7.5

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

   if -2.153952848057346e-251 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 55.8

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

   2. Taylor expanded around inf 22.3

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.153952848057346 \cdot 10^{-251} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;x + \left(\frac{z}{a - t} \cdot \left(y - x\right) + \left(-\frac{t}{a - t}\right) \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))