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

Average Error: 24.8 → 8.7

Time: 20.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.728574168981643158068947885483381777474 \cdot 10^{-279} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0\right):\\ \;\;\;\;x + \left(\left(\sqrt[3]{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)\right)} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)}\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r441580 = x;
        double r441581 = y;
        double r441582 = z;
        double r441583 = r441581 - r441582;
        double r441584 = t;
        double r441585 = r441584 - r441580;
        double r441586 = r441583 * r441585;
        double r441587 = a;
        double r441588 = r441587 - r441582;
        double r441589 = r441586 / r441588;
        double r441590 = r441580 + r441589;
        return r441590;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r441591 = x;
        double r441592 = y;
        double r441593 = z;
        double r441594 = r441592 - r441593;
        double r441595 = t;
        double r441596 = r441595 - r441591;
        double r441597 = r441594 * r441596;
        double r441598 = a;
        double r441599 = r441598 - r441593;
        double r441600 = r441597 / r441599;
        double r441601 = r441591 + r441600;
        double r441602 = -1.7285741689816432e-279;
        bool r441603 = r441601 <= r441602;
        double r441604 = 0.0;
        bool r441605 = r441601 <= r441604;
        double r441606 = !r441605;
        bool r441607 = r441603 || r441606;
        double r441608 = cbrt(r441594);
        double r441609 = r441608 * r441608;
        double r441610 = cbrt(r441599);
        double r441611 = r441609 / r441610;
        double r441612 = r441608 / r441610;
        double r441613 = cbrt(r441596);
        double r441614 = r441613 * r441613;
        double r441615 = r441612 * r441614;
        double r441616 = r441611 * r441615;
        double r441617 = cbrt(r441616);
        double r441618 = r441610 * r441610;
        double r441619 = r441594 / r441618;
        double r441620 = r441619 * r441614;
        double r441621 = cbrt(r441620);
        double r441622 = r441617 * r441621;
        double r441623 = r441622 * r441621;
        double r441624 = r441613 / r441610;
        double r441625 = r441623 * r441624;
        double r441626 = r441591 + r441625;
        double r441627 = r441591 * r441592;
        double r441628 = r441627 / r441593;
        double r441629 = r441628 + r441595;
        double r441630 = r441595 * r441592;
        double r441631 = r441630 / r441593;
        double r441632 = r441629 - r441631;
        double r441633 = r441607 ? r441626 : r441632;
        return r441633;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	11.9
Herbie	8.7

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -1.7285741689816432e-279 or 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 21.6

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

   3. Applied add-cube-cbrt 22.1

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

   4. Applied times-frac 8.2

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 8.2

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{\color{blue}{1 \cdot \left(a - z\right)}}}\]

   7. Applied cbrt-prod 8.2

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{a - z}}}\]

   8. Applied add-cube-cbrt 8.4

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\color{blue}{\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \sqrt[3]{t - x}}}{\sqrt[3]{1} \cdot \sqrt[3]{a - z}}\]

   9. Applied times-frac 8.4

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \color{blue}{\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\right)}\]

   10. Applied associate-*r* 7.7

       \[\leadsto x + \color{blue}{\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}}\]

   11. Simplified 7.7

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

   13. Applied add-cube-cbrt 7.9

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

   15. Applied add-cube-cbrt 7.8

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

   16. Applied times-frac 7.8

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

   17. Applied associate-*l* 7.8

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

   if -1.7285741689816432e-279 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program 59.7

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

   2. Taylor expanded around inf 18.8

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.728574168981643158068947885483381777474 \cdot 10^{-279} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0\right):\\ \;\;\;\;x + \left(\left(\sqrt[3]{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)\right)} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)}\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.25361310560950359e188) (- t (* (/ y z) (- t x))) (if (< z 4.44670236911381103e64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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