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

Average Error: 24.4 → 11.0

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.955121187742057667599181426946027509407 \cdot 10^{-190}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}\right) \cdot \sqrt[3]{\sqrt[3]{a - z}}\right)} \cdot \frac{t - x}{1 \cdot \sqrt[3]{a - z}}\\ \mathbf{elif}\;a \le 6.559704169034857271833525661466055522025 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}\right) \cdot \sqrt[3]{\sqrt[3]{a - z}}\right)} \cdot \frac{t - x}{1 \cdot \left(\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r827441 = x;
        double r827442 = y;
        double r827443 = z;
        double r827444 = r827442 - r827443;
        double r827445 = t;
        double r827446 = r827445 - r827441;
        double r827447 = r827444 * r827446;
        double r827448 = a;
        double r827449 = r827448 - r827443;
        double r827450 = r827447 / r827449;
        double r827451 = r827441 + r827450;
        return r827451;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r827452 = a;
        double r827453 = -7.955121187742058e-190;
        bool r827454 = r827452 <= r827453;
        double r827455 = x;
        double r827456 = y;
        double r827457 = z;
        double r827458 = r827456 - r827457;
        double r827459 = r827452 - r827457;
        double r827460 = cbrt(r827459);
        double r827461 = cbrt(r827460);
        double r827462 = r827461 * r827461;
        double r827463 = r827462 * r827461;
        double r827464 = r827460 * r827463;
        double r827465 = r827458 / r827464;
        double r827466 = t;
        double r827467 = r827466 - r827455;
        double r827468 = 1.0;
        double r827469 = r827468 * r827460;
        double r827470 = r827467 / r827469;
        double r827471 = r827465 * r827470;
        double r827472 = r827455 + r827471;
        double r827473 = 6.559704169034857e-143;
        bool r827474 = r827452 <= r827473;
        double r827475 = r827455 * r827456;
        double r827476 = r827475 / r827457;
        double r827477 = r827476 + r827466;
        double r827478 = r827466 * r827456;
        double r827479 = r827478 / r827457;
        double r827480 = r827477 - r827479;
        double r827481 = r827460 * r827460;
        double r827482 = cbrt(r827481);
        double r827483 = r827482 * r827461;
        double r827484 = r827468 * r827483;
        double r827485 = r827467 / r827484;
        double r827486 = r827465 * r827485;
        double r827487 = r827455 + r827486;
        double r827488 = r827474 ? r827480 : r827487;
        double r827489 = r827454 ? r827472 : r827488;
        return r827489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.4
Target	11.9
Herbie	11.0

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

Derivation

1. Split input into 3 regimes

2. if a < -7.955121187742058e-190

   1. Initial program 22.3

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

   3. Applied add-cube-cbrt 22.7

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

   4. Applied times-frac 10.6

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

   6. Applied *-un-lft-identity 10.6

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

   7. Applied cbrt-prod 10.6

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

   8. Simplified 10.6

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

   10. Applied add-cube-cbrt 10.8

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

   if -7.955121187742058e-190 < a < 6.559704169034857e-143

   1. Initial program 30.8

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

   2. Taylor expanded around inf 12.8

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

   if 6.559704169034857e-143 < a

   1. Initial program 23.3

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

   3. Applied add-cube-cbrt 23.7

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

   4. Applied times-frac 10.1

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

   6. Applied *-un-lft-identity 10.1

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

   7. Applied cbrt-prod 10.1

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

   8. Simplified 10.1

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

   10. Applied add-cube-cbrt 10.3

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

   12. Applied add-cube-cbrt 10.3

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

   13. Applied cbrt-prod 10.4

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.955121187742057667599181426946027509407 \cdot 10^{-190}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}\right) \cdot \sqrt[3]{\sqrt[3]{a - z}}\right)} \cdot \frac{t - x}{1 \cdot \sqrt[3]{a - z}}\\ \mathbf{elif}\;a \le 6.559704169034857271833525661466055522025 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}\right) \cdot \sqrt[3]{\sqrt[3]{a - z}}\right)} \cdot \frac{t - x}{1 \cdot \left(\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))