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

Average Error: 24.8 → 8.3

Time: 53.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} = -\infty:\\ \;\;\;\;\frac{y - z}{\frac{a - z}{t - x}} + x\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -3.344938911954521452455462044931109127167 \cdot 10^{-159}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\ \;\;\;\;x + \frac{t - x}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}} \cdot \frac{\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}} \cdot \frac{\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r31049659 = x;
        double r31049660 = y;
        double r31049661 = z;
        double r31049662 = r31049660 - r31049661;
        double r31049663 = t;
        double r31049664 = r31049663 - r31049659;
        double r31049665 = r31049662 * r31049664;
        double r31049666 = a;
        double r31049667 = r31049666 - r31049661;
        double r31049668 = r31049665 / r31049667;
        double r31049669 = r31049659 + r31049668;
        return r31049669;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r31049670 = x;
        double r31049671 = y;
        double r31049672 = z;
        double r31049673 = r31049671 - r31049672;
        double r31049674 = t;
        double r31049675 = r31049674 - r31049670;
        double r31049676 = r31049673 * r31049675;
        double r31049677 = a;
        double r31049678 = r31049677 - r31049672;
        double r31049679 = r31049676 / r31049678;
        double r31049680 = r31049670 + r31049679;
        double r31049681 = -inf.0;
        bool r31049682 = r31049680 <= r31049681;
        double r31049683 = r31049678 / r31049675;
        double r31049684 = r31049673 / r31049683;
        double r31049685 = r31049684 + r31049670;
        double r31049686 = -3.3449389119545215e-159;
        bool r31049687 = r31049680 <= r31049686;
        double r31049688 = -1.5335432679341662e-301;
        bool r31049689 = r31049680 <= r31049688;
        double r31049690 = cbrt(r31049678);
        double r31049691 = cbrt(r31049690);
        double r31049692 = cbrt(r31049691);
        double r31049693 = r31049675 / r31049692;
        double r31049694 = r31049690 * r31049690;
        double r31049695 = r31049673 / r31049694;
        double r31049696 = cbrt(r31049694);
        double r31049697 = r31049695 / r31049696;
        double r31049698 = r31049692 * r31049692;
        double r31049699 = r31049697 / r31049698;
        double r31049700 = r31049693 * r31049699;
        double r31049701 = r31049670 + r31049700;
        double r31049702 = 0.0;
        bool r31049703 = r31049680 <= r31049702;
        double r31049704 = r31049670 * r31049671;
        double r31049705 = r31049704 / r31049672;
        double r31049706 = r31049705 + r31049674;
        double r31049707 = r31049671 * r31049674;
        double r31049708 = r31049707 / r31049672;
        double r31049709 = r31049706 - r31049708;
        double r31049710 = r31049703 ? r31049709 : r31049701;
        double r31049711 = r31049689 ? r31049701 : r31049710;
        double r31049712 = r31049687 ? r31049680 : r31049711;
        double r31049713 = r31049682 ? r31049685 : r31049712;
        return r31049713;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	11.8
Herbie	8.3

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

Derivation

1. Split input into 4 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l* 17.4

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

   if -inf.0 < (+ x (/ (* (- y z) (- t x)) (- a z))) < -3.3449389119545215e-159

   1. Initial program 1.5

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

   3. Applied add-cube-cbrt 2.2

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

   4. Applied times-frac 3.5

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

   6. Applied *-un-lft-identity 3.5

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

   7. Applied associate-*l* 3.5

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

   8. Simplified 1.5

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

   if -3.3449389119545215e-159 < (+ x (/ (* (- y z) (- t x)) (- a z))) < -1.5335432679341662e-301 or 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 20.8

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

   3. Applied add-cube-cbrt 21.3

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

   4. Applied times-frac 8.0

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

   6. Applied add-cube-cbrt 8.1

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

   7. Applied cbrt-prod 8.1

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

   8. Applied *-un-lft-identity 8.1

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

   9. Applied times-frac 8.1

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

   10. Applied associate-*r* 7.8

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

   11. Simplified 7.8

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

   13. Applied add-cube-cbrt 8.0

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

   14. Applied *-un-lft-identity 8.0

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

   15. Applied times-frac 8.0

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

   16. Applied associate-*r* 8.0

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

   17. Simplified 8.0

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

   if -1.5335432679341662e-301 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program 60.6

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

   2. Taylor expanded around inf 17.8

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} = -\infty:\\ \;\;\;\;\frac{y - z}{\frac{a - z}{t - x}} + x\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -3.344938911954521452455462044931109127167 \cdot 10^{-159}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -1.533543267934166183549967405153551686617 \cdot 10^{-301}:\\ \;\;\;\;x + \frac{t - x}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}} \cdot \frac{\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 0.0:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - x}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}} \cdot \frac{\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))