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

Average Error: 24.2 → 8.9

Time: 17.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -4.82516022252417878 \cdot 10^{-304}:\\ \;\;\;\;x + \left(\left(\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 8.72300029818557341 \cdot 10^{284}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r545769 = x;
        double r545770 = y;
        double r545771 = z;
        double r545772 = r545770 - r545771;
        double r545773 = t;
        double r545774 = r545773 - r545769;
        double r545775 = r545772 * r545774;
        double r545776 = a;
        double r545777 = r545776 - r545771;
        double r545778 = r545775 / r545777;
        double r545779 = r545769 + r545778;
        return r545779;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r545780 = x;
        double r545781 = y;
        double r545782 = z;
        double r545783 = r545781 - r545782;
        double r545784 = t;
        double r545785 = r545784 - r545780;
        double r545786 = r545783 * r545785;
        double r545787 = a;
        double r545788 = r545787 - r545782;
        double r545789 = r545786 / r545788;
        double r545790 = r545780 + r545789;
        double r545791 = -4.825160222524179e-304;
        bool r545792 = r545790 <= r545791;
        double r545793 = cbrt(r545785);
        double r545794 = r545793 * r545793;
        double r545795 = cbrt(r545788);
        double r545796 = r545795 * r545795;
        double r545797 = r545783 / r545796;
        double r545798 = cbrt(r545797);
        double r545799 = r545798 * r545798;
        double r545800 = r545794 * r545799;
        double r545801 = r545800 * r545798;
        double r545802 = r545793 / r545795;
        double r545803 = r545801 * r545802;
        double r545804 = r545780 + r545803;
        double r545805 = 0.0;
        bool r545806 = r545790 <= r545805;
        double r545807 = r545780 * r545781;
        double r545808 = r545807 / r545782;
        double r545809 = r545808 + r545784;
        double r545810 = r545784 * r545781;
        double r545811 = r545810 / r545782;
        double r545812 = r545809 - r545811;
        double r545813 = 8.723000298185573e+284;
        bool r545814 = r545790 <= r545813;
        double r545815 = 1.0;
        double r545816 = r545815 / r545788;
        double r545817 = r545786 * r545816;
        double r545818 = r545780 + r545817;
        double r545819 = r545785 / r545788;
        double r545820 = r545783 * r545819;
        double r545821 = r545780 + r545820;
        double r545822 = r545814 ? r545818 : r545821;
        double r545823 = r545806 ? r545812 : r545822;
        double r545824 = r545792 ? r545804 : r545823;
        return r545824;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.2
Target	11.7
Herbie	8.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -4.825160222524179e-304

   1. Initial program 21.1

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

   3. Applied add-cube-cbrt 21.6

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

   4. Applied times-frac 8.7

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

   6. Applied *-un-lft-identity 8.7

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

   7. Applied cbrt-prod 8.7

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

   8. Applied add-cube-cbrt 8.9

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

   9. Applied times-frac 8.9

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

   10. Applied associate-*r* 8.2

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

   11. Simplified 8.2

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

   13. Applied add-cube-cbrt 8.3

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

   14. Applied associate-*r* 8.3

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

   if -4.825160222524179e-304 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program 60.8

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

   2. Taylor expanded around inf 19.3

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

   if 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 8.723000298185573e+284

   1. Initial program 2.4

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

   3. Applied div-inv 2.5

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

   if 8.723000298185573e+284 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 59.7

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

   3. Applied *-un-lft-identity 59.7

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

   4. Applied times-frac 18.3

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

   5. Simplified 18.3

      \[\leadsto x + \color{blue}{\left(y - z\right)} \cdot \frac{t - x}{a - z}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -4.82516022252417878 \cdot 10^{-304}:\\ \;\;\;\;x + \left(\left(\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 8.72300029818557341 \cdot 10^{284}:\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \left(t - x\right)\right) \cdot \frac{1}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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