Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.7 → 7.6

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -3.3728247353674194 \cdot 10^{-295} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 0.0\right):\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r127631 = x;
        double r127632 = y;
        double r127633 = z;
        double r127634 = r127632 - r127633;
        double r127635 = t;
        double r127636 = r127635 - r127631;
        double r127637 = a;
        double r127638 = r127637 - r127633;
        double r127639 = r127636 / r127638;
        double r127640 = r127634 * r127639;
        double r127641 = r127631 + r127640;
        return r127641;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r127642 = x;
        double r127643 = y;
        double r127644 = z;
        double r127645 = r127643 - r127644;
        double r127646 = t;
        double r127647 = r127646 - r127642;
        double r127648 = a;
        double r127649 = r127648 - r127644;
        double r127650 = r127647 / r127649;
        double r127651 = r127645 * r127650;
        double r127652 = r127642 + r127651;
        double r127653 = -3.3728247353674194e-295;
        bool r127654 = r127652 <= r127653;
        double r127655 = 0.0;
        bool r127656 = r127652 <= r127655;
        double r127657 = !r127656;
        bool r127658 = r127654 || r127657;
        double r127659 = cbrt(r127645);
        double r127660 = r127659 * r127659;
        double r127661 = cbrt(r127649);
        double r127662 = r127660 / r127661;
        double r127663 = cbrt(r127661);
        double r127664 = r127662 / r127663;
        double r127665 = r127659 / r127661;
        double r127666 = r127665 / r127663;
        double r127667 = r127647 / r127663;
        double r127668 = r127666 * r127667;
        double r127669 = r127664 * r127668;
        double r127670 = r127642 + r127669;
        double r127671 = r127642 * r127643;
        double r127672 = r127671 / r127644;
        double r127673 = r127672 + r127646;
        double r127674 = r127646 * r127643;
        double r127675 = r127674 / r127644;
        double r127676 = r127673 - r127675;
        double r127677 = r127658 ? r127670 : r127676;
        return r127677;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if (+ x (* (- y z) (/ (- t x) (- a z)))) < -3.3728247353674194e-295 or 0.0 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt 7.9

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

   4. Applied *-un-lft-identity 7.9

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

   5. Applied times-frac 7.9

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

   6. Applied associate-*r* 5.3

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

   7. Simplified 5.3

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

   9. Applied add-cube-cbrt 5.5

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

   10. Applied *-un-lft-identity 5.5

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

   11. Applied times-frac 5.5

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

   12. Applied associate-*r* 5.3

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

   13. Simplified 5.3

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

   15. Applied add-cube-cbrt 5.2

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

   16. Applied times-frac 5.2

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

   17. Applied times-frac 5.2

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

   18. Applied associate-*l* 4.7

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

   if -3.3728247353674194e-295 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 0.0

   1. Initial program 61.4

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

   2. Taylor expanded around inf 25.7

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -3.3728247353674194 \cdot 10^{-295} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 0.0\right):\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))