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

Average Error: 24.7 → 10.5

Time: 24.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le \frac{-8288978770693565}{1.886981212410770676120777290494134445458 \cdot 10^{168}} \lor \neg \left(a \le \frac{3670337189150779}{2.772669694120814859578414184143083703436 \cdot 10^{129}}\right):\\ \;\;\;\;x + \left(\left(\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r541608 = x;
        double r541609 = y;
        double r541610 = z;
        double r541611 = r541609 - r541610;
        double r541612 = t;
        double r541613 = r541612 - r541608;
        double r541614 = r541611 * r541613;
        double r541615 = a;
        double r541616 = r541615 - r541610;
        double r541617 = r541614 / r541616;
        double r541618 = r541608 + r541617;
        return r541618;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r541619 = a;
        double r541620 = -8288978770693565.0;
        double r541621 = 1.8869812124107707e+168;
        double r541622 = r541620 / r541621;
        bool r541623 = r541619 <= r541622;
        double r541624 = 3670337189150779.0;
        double r541625 = 2.772669694120815e+129;
        double r541626 = r541624 / r541625;
        bool r541627 = r541619 <= r541626;
        double r541628 = !r541627;
        bool r541629 = r541623 || r541628;
        double r541630 = x;
        double r541631 = y;
        double r541632 = z;
        double r541633 = r541631 - r541632;
        double r541634 = cbrt(r541633);
        double r541635 = r541634 * r541634;
        double r541636 = r541619 - r541632;
        double r541637 = cbrt(r541636);
        double r541638 = r541635 / r541637;
        double r541639 = r541634 / r541637;
        double r541640 = t;
        double r541641 = r541640 - r541630;
        double r541642 = cbrt(r541641);
        double r541643 = r541637 * r541637;
        double r541644 = cbrt(r541643);
        double r541645 = cbrt(r541644);
        double r541646 = r541645 * r541645;
        double r541647 = r541642 / r541646;
        double r541648 = r541639 * r541647;
        double r541649 = r541638 * r541648;
        double r541650 = r541642 / r541645;
        double r541651 = r541649 * r541650;
        double r541652 = cbrt(r541637);
        double r541653 = r541642 / r541652;
        double r541654 = r541651 * r541653;
        double r541655 = r541630 + r541654;
        double r541656 = r541630 * r541631;
        double r541657 = r541656 / r541632;
        double r541658 = r541657 + r541640;
        double r541659 = r541640 * r541631;
        double r541660 = r541659 / r541632;
        double r541661 = r541658 - r541660;
        double r541662 = r541629 ? r541655 : r541661;
        return r541662;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.7
Target	11.9
Herbie	10.5

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

Derivation

1. Split input into 2 regimes

2. if a < -4.3927192895067175e-153 or 1.323755655761443e-114 < a

   1. Initial program 22.9

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

   3. Applied add-cube-cbrt 23.3

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

   4. Applied times-frac 9.6

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

   6. Applied add-cube-cbrt 9.7

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

   7. Applied cbrt-prod 9.7

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

   8. Applied add-cube-cbrt 9.8

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

   9. Applied times-frac 9.8

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

   10. Applied associate-*r* 9.4

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

   12. Applied add-cube-cbrt 9.5

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

   13. Applied times-frac 9.5

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

   14. Applied associate-*r* 9.5

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

   16. Applied add-cube-cbrt 9.5

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

   17. Applied times-frac 9.5

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

   18. Applied associate-*l* 9.4

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

   if -4.3927192895067175e-153 < a < 1.323755655761443e-114

   1. Initial program 30.0

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

   2. Taylor expanded around inf 13.6

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le \frac{-8288978770693565}{1.886981212410770676120777290494134445458 \cdot 10^{168}} \lor \neg \left(a \le \frac{3670337189150779}{2.772669694120814859578414184143083703436 \cdot 10^{129}}\right):\\ \;\;\;\;x + \left(\left(\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))