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

Average Error: 24.5 → 8.5

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.20080531218694779373582477752182757726 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0\right):\\ \;\;\;\;x + \left(\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r478760 = x;
        double r478761 = y;
        double r478762 = z;
        double r478763 = r478761 - r478762;
        double r478764 = t;
        double r478765 = r478764 - r478760;
        double r478766 = r478763 * r478765;
        double r478767 = a;
        double r478768 = r478767 - r478762;
        double r478769 = r478766 / r478768;
        double r478770 = r478760 + r478769;
        return r478770;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r478771 = x;
        double r478772 = y;
        double r478773 = z;
        double r478774 = r478772 - r478773;
        double r478775 = t;
        double r478776 = r478775 - r478771;
        double r478777 = r478774 * r478776;
        double r478778 = a;
        double r478779 = r478778 - r478773;
        double r478780 = r478777 / r478779;
        double r478781 = r478771 + r478780;
        double r478782 = -1.2008053121869478e-290;
        bool r478783 = r478781 <= r478782;
        double r478784 = 0.0;
        bool r478785 = r478781 <= r478784;
        double r478786 = !r478785;
        bool r478787 = r478783 || r478786;
        double r478788 = cbrt(r478774);
        double r478789 = r478788 * r478788;
        double r478790 = cbrt(r478779);
        double r478791 = r478789 / r478790;
        double r478792 = r478788 / r478790;
        double r478793 = cbrt(r478776);
        double r478794 = r478793 * r478793;
        double r478795 = r478792 * r478794;
        double r478796 = r478791 * r478795;
        double r478797 = r478793 / r478790;
        double r478798 = r478796 * r478797;
        double r478799 = r478771 + r478798;
        double r478800 = r478771 * r478772;
        double r478801 = r478800 / r478773;
        double r478802 = r478801 + r478775;
        double r478803 = r478775 * r478772;
        double r478804 = r478803 / r478773;
        double r478805 = r478802 - r478804;
        double r478806 = r478787 ? r478799 : r478805;
        return r478806;
}

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	12.0
Herbie	8.5

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -1.2008053121869478e-290 or 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 21.2

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

   3. Applied add-cube-cbrt 21.7

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

   4. Applied times-frac 8.4

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

   6. Applied *-un-lft-identity 8.4

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

   7. Applied cbrt-prod 8.4

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

   8. Applied add-cube-cbrt 8.5

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

   9. Applied times-frac 8.5

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

   10. Applied associate-*r* 7.9

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

   11. Simplified 7.9

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

   13. Applied add-cube-cbrt 7.9

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

   14. Applied times-frac 7.9

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

   15. Applied associate-*l* 7.5

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

   if -1.2008053121869478e-290 < (+ 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 19.0

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.20080531218694779373582477752182757726 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0\right):\\ \;\;\;\;x + \left(\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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))))