Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.7 → 6.9

Time: 22.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le -1.3435650229451346 \cdot 10^{-276}:\\ \;\;\;\;x + \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}} \cdot \frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}}\\ \mathbf{elif}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le 0.0:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\frac{t - x}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r7240726 = x;
        double r7240727 = y;
        double r7240728 = z;
        double r7240729 = r7240727 - r7240728;
        double r7240730 = t;
        double r7240731 = r7240730 - r7240726;
        double r7240732 = a;
        double r7240733 = r7240732 - r7240728;
        double r7240734 = r7240731 / r7240733;
        double r7240735 = r7240729 * r7240734;
        double r7240736 = r7240726 + r7240735;
        return r7240736;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r7240737 = x;
        double r7240738 = t;
        double r7240739 = r7240738 - r7240737;
        double r7240740 = a;
        double r7240741 = z;
        double r7240742 = r7240740 - r7240741;
        double r7240743 = r7240739 / r7240742;
        double r7240744 = y;
        double r7240745 = r7240744 - r7240741;
        double r7240746 = r7240743 * r7240745;
        double r7240747 = r7240737 + r7240746;
        double r7240748 = -1.3435650229451346e-276;
        bool r7240749 = r7240747 <= r7240748;
        double r7240750 = cbrt(r7240739);
        double r7240751 = cbrt(r7240742);
        double r7240752 = r7240750 / r7240751;
        double r7240753 = r7240751 / r7240750;
        double r7240754 = r7240753 * r7240753;
        double r7240755 = r7240745 / r7240754;
        double r7240756 = r7240752 * r7240755;
        double r7240757 = r7240737 + r7240756;
        double r7240758 = 0.0;
        bool r7240759 = r7240747 <= r7240758;
        double r7240760 = r7240737 / r7240741;
        double r7240761 = r7240738 / r7240741;
        double r7240762 = r7240760 - r7240761;
        double r7240763 = r7240744 * r7240762;
        double r7240764 = r7240763 + r7240738;
        double r7240765 = r7240739 / r7240751;
        double r7240766 = r7240751 * r7240751;
        double r7240767 = r7240745 / r7240766;
        double r7240768 = r7240765 * r7240767;
        double r7240769 = r7240768 + r7240737;
        double r7240770 = r7240759 ? r7240764 : r7240769;
        double r7240771 = r7240749 ? r7240757 : r7240770;
        return r7240771;
}

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 3 regimes

2. if (+ x (* (- y z) (/ (- t x) (- a z)))) < -1.3435650229451346e-276

   1. Initial program 6.9

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

   3. Applied add-cube-cbrt 7.6

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

   4. Applied *-un-lft-identity 7.6

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

   5. Applied times-frac 7.6

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

   6. Applied associate-*r* 5.0

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

   7. Simplified 5.0

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

   9. Applied *-un-lft-identity 5.0

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

   10. Applied cbrt-prod 5.0

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

   11. Applied add-cube-cbrt 5.2

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

   12. Applied times-frac 5.2

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

   13. Applied associate-*r* 4.6

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

   14. Simplified 4.6

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

   if -1.3435650229451346e-276 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 0.0

   1. Initial program 60.6

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

   3. Applied add-cube-cbrt 60.5

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

   4. Applied *-un-lft-identity 60.5

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

   5. Applied times-frac 60.4

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

   6. Applied associate-*r* 59.3

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

   7. Simplified 59.3

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

   9. Applied *-un-lft-identity 59.3

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

   10. Applied cbrt-prod 59.3

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

   11. Applied add-cube-cbrt 59.2

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

   12. Applied times-frac 59.3

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

   13. Applied associate-*r* 58.8

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

   14. Simplified 59.4

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

   15. Taylor expanded around inf 24.7

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

   16. Simplified 20.3

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

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

   1. Initial program 7.1

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

   3. Applied add-cube-cbrt 7.8

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

   4. Applied *-un-lft-identity 7.8

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

   5. Applied times-frac 7.8

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

   6. Applied associate-*r* 4.8

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

   7. Simplified 4.8

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

   9. Applied *-commutative 4.8

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le -1.3435650229451346 \cdot 10^{-276}:\\ \;\;\;\;x + \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}} \cdot \frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}}\\ \mathbf{elif}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le 0.0:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\frac{t - x}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  (+ x (* (- y z) (/ (- t x) (- a z)))))