Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 15.0 → 7.2

Time: 6.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -3.294617881327087 \cdot 10^{-290} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 0.0\right):\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\left(\sqrt[3]{\sqrt[3]{y - z}} \cdot \sqrt[3]{\sqrt[3]{y - z}}\right) \cdot \sqrt[3]{\sqrt[3]{y - z}}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r183197 = x;
        double r183198 = y;
        double r183199 = z;
        double r183200 = r183198 - r183199;
        double r183201 = t;
        double r183202 = r183201 - r183197;
        double r183203 = a;
        double r183204 = r183203 - r183199;
        double r183205 = r183202 / r183204;
        double r183206 = r183200 * r183205;
        double r183207 = r183197 + r183206;
        return r183207;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r183208 = x;
        double r183209 = y;
        double r183210 = z;
        double r183211 = r183209 - r183210;
        double r183212 = t;
        double r183213 = r183212 - r183208;
        double r183214 = a;
        double r183215 = r183214 - r183210;
        double r183216 = r183213 / r183215;
        double r183217 = r183211 * r183216;
        double r183218 = r183208 + r183217;
        double r183219 = -3.294617881327087e-290;
        bool r183220 = r183218 <= r183219;
        double r183221 = 0.0;
        bool r183222 = r183218 <= r183221;
        double r183223 = !r183222;
        bool r183224 = r183220 || r183223;
        double r183225 = cbrt(r183211);
        double r183226 = r183225 * r183225;
        double r183227 = cbrt(r183215);
        double r183228 = r183226 / r183227;
        double r183229 = cbrt(r183225);
        double r183230 = r183229 * r183229;
        double r183231 = r183230 * r183229;
        double r183232 = r183231 / r183227;
        double r183233 = r183213 / r183227;
        double r183234 = r183232 * r183233;
        double r183235 = r183228 * r183234;
        double r183236 = r183208 + r183235;
        double r183237 = r183208 / r183210;
        double r183238 = r183212 / r183210;
        double r183239 = r183237 - r183238;
        double r183240 = r183209 * r183239;
        double r183241 = r183240 + r183212;
        double r183242 = r183224 ? r183236 : r183241;
        return r183242;
}

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)))) < -3.294617881327087e-290 or 0.0 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 8.0

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

   4. Applied *-un-lft-identity 8.0

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

   5. Applied times-frac 8.0

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

   6. Applied associate-*r* 5.1

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

   7. Simplified 5.1

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

   9. Applied add-cube-cbrt 4.9

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

   10. Applied times-frac 5.0

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

   11. Applied associate-*l* 4.6

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

   13. Applied add-cube-cbrt 5.0

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

   if -3.294617881327087e-290 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 0.0

   1. Initial program 61.0

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

   3. Applied add-cube-cbrt 60.8

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

   4. Applied *-un-lft-identity 60.8

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

   5. Applied times-frac 60.8

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

   6. Applied associate-*r* 60.3

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

   7. Simplified 60.3

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

   9. Applied add-cube-cbrt 60.5

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

   10. Applied times-frac 60.4

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

   11. Applied associate-*l* 60.4

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

   12. Taylor expanded around inf 26.0

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

   13. Simplified 20.7

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -3.294617881327087 \cdot 10^{-290} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 0.0\right):\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\left(\sqrt[3]{\sqrt[3]{y - z}} \cdot \sqrt[3]{\sqrt[3]{y - z}}\right) \cdot \sqrt[3]{\sqrt[3]{y - z}}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))))