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

Average Error: 24.5 → 9.1

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.153952848057346 \cdot 10^{-251} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\left(x + \frac{z}{a - t} \cdot \left(y - x\right)\right) + \left(y - x\right) \cdot \left(-\frac{\frac{t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r633618 = x;
        double r633619 = y;
        double r633620 = r633619 - r633618;
        double r633621 = z;
        double r633622 = t;
        double r633623 = r633621 - r633622;
        double r633624 = r633620 * r633623;
        double r633625 = a;
        double r633626 = r633625 - r633622;
        double r633627 = r633624 / r633626;
        double r633628 = r633618 + r633627;
        return r633628;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r633629 = x;
        double r633630 = y;
        double r633631 = r633630 - r633629;
        double r633632 = z;
        double r633633 = t;
        double r633634 = r633632 - r633633;
        double r633635 = r633631 * r633634;
        double r633636 = a;
        double r633637 = r633636 - r633633;
        double r633638 = r633635 / r633637;
        double r633639 = r633629 + r633638;
        double r633640 = -2.153952848057346e-251;
        bool r633641 = r633639 <= r633640;
        double r633642 = 0.0;
        bool r633643 = r633639 <= r633642;
        double r633644 = !r633643;
        bool r633645 = r633641 || r633644;
        double r633646 = r633632 / r633637;
        double r633647 = r633646 * r633631;
        double r633648 = r633629 + r633647;
        double r633649 = cbrt(r633637);
        double r633650 = r633649 * r633649;
        double r633651 = r633633 / r633650;
        double r633652 = r633651 / r633649;
        double r633653 = -r633652;
        double r633654 = r633631 * r633653;
        double r633655 = r633648 + r633654;
        double r633656 = r633629 * r633632;
        double r633657 = r633656 / r633633;
        double r633658 = r633630 + r633657;
        double r633659 = r633632 * r633630;
        double r633660 = r633659 / r633633;
        double r633661 = r633658 - r633660;
        double r633662 = r633645 ? r633655 : r633661;
        return r633662;
}

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	9.1

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

Derivation

1. Split input into 2 regimes

2. if (+ 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. Applied associate-+r+ 7.5

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

   12. Simplified 7.5

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

   14. Applied add-cube-cbrt 7.8

       \[\leadsto \left(x + \frac{z}{a - t} \cdot \left(y - x\right)\right) + \left(y - x\right) \cdot \left(-\frac{t}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\right)\]

   15. Applied associate-/r* 7.8

       \[\leadsto \left(x + \frac{z}{a - t} \cdot \left(y - x\right)\right) + \left(y - x\right) \cdot \left(-\color{blue}{\frac{\frac{t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}}\right)\]

   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 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.153952848057346 \cdot 10^{-251} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\left(x + \frac{z}{a - t} \cdot \left(y - x\right)\right) + \left(y - x\right) \cdot \left(-\frac{\frac{t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

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))))