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

Average Error: 24.1 → 10.0

Time: 26.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.16213479572799396537678486310634204319 \cdot 10^{-127}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{t - x}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}\right)\\ \mathbf{elif}\;a \le 3.08635240287905078638271475344473661072 \cdot 10^{-58}:\\ \;\;\;\;\left(\frac{x}{z} - \frac{t}{z}\right) \cdot y + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right)} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r32406326 = x;
        double r32406327 = y;
        double r32406328 = z;
        double r32406329 = r32406327 - r32406328;
        double r32406330 = t;
        double r32406331 = r32406330 - r32406326;
        double r32406332 = r32406329 * r32406331;
        double r32406333 = a;
        double r32406334 = r32406333 - r32406328;
        double r32406335 = r32406332 / r32406334;
        double r32406336 = r32406326 + r32406335;
        return r32406336;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r32406337 = a;
        double r32406338 = -6.162134795727994e-127;
        bool r32406339 = r32406337 <= r32406338;
        double r32406340 = x;
        double r32406341 = y;
        double r32406342 = z;
        double r32406343 = r32406341 - r32406342;
        double r32406344 = cbrt(r32406343);
        double r32406345 = r32406344 * r32406344;
        double r32406346 = r32406337 - r32406342;
        double r32406347 = cbrt(r32406346);
        double r32406348 = r32406345 / r32406347;
        double r32406349 = t;
        double r32406350 = r32406349 - r32406340;
        double r32406351 = r32406350 / r32406347;
        double r32406352 = r32406344 / r32406347;
        double r32406353 = r32406351 * r32406352;
        double r32406354 = r32406348 * r32406353;
        double r32406355 = r32406340 + r32406354;
        double r32406356 = 3.0863524028790508e-58;
        bool r32406357 = r32406337 <= r32406356;
        double r32406358 = r32406340 / r32406342;
        double r32406359 = r32406349 / r32406342;
        double r32406360 = r32406358 - r32406359;
        double r32406361 = r32406360 * r32406341;
        double r32406362 = r32406361 + r32406349;
        double r32406363 = r32406347 * r32406347;
        double r32406364 = cbrt(r32406363);
        double r32406365 = r32406364 * r32406363;
        double r32406366 = r32406343 / r32406365;
        double r32406367 = cbrt(r32406347);
        double r32406368 = r32406350 / r32406367;
        double r32406369 = r32406366 * r32406368;
        double r32406370 = r32406340 + r32406369;
        double r32406371 = r32406357 ? r32406362 : r32406370;
        double r32406372 = r32406339 ? r32406355 : r32406371;
        return r32406372;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.1
Target	11.9
Herbie	10.0

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

Derivation

1. Split input into 3 regimes

2. if a < -6.162134795727994e-127

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.9

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

   4. Applied times-frac 8.3

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

   6. Applied add-cube-cbrt 8.3

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

   7. Applied times-frac 8.3

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

   8. Applied associate-*l* 8.1

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

   if -6.162134795727994e-127 < a < 3.0863524028790508e-58

   1. Initial program 29.0

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

   3. Applied add-cube-cbrt 29.6

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

   4. Applied times-frac 21.5

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

   6. Applied add-cube-cbrt 21.2

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

   7. Applied times-frac 21.2

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

   8. Applied associate-*l* 20.7

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

   9. Taylor expanded around inf 16.4

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

   10. Simplified 13.5

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

   if 3.0863524028790508e-58 < a

   1. Initial program 22.2

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

   3. Applied add-cube-cbrt 22.6

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

   4. Applied times-frac 8.6

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

   6. Applied add-cube-cbrt 8.6

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

   7. Applied cbrt-prod 8.7

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

   8. Applied *-un-lft-identity 8.7

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

   9. Applied times-frac 8.7

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

   10. Applied associate-*r* 8.6

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

   11. Simplified 8.6

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.16213479572799396537678486310634204319 \cdot 10^{-127}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{t - x}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}\right)\\ \mathbf{elif}\;a \le 3.08635240287905078638271475344473661072 \cdot 10^{-58}:\\ \;\;\;\;\left(\frac{x}{z} - \frac{t}{z}\right) \cdot y + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right)} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :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))))