Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.7 → 10.3

Time: 49.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.016722143479182721107069384915645128898 \cdot 10^{-79}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\frac{\sqrt[3]{a - z}}{t - x}}\\ \mathbf{elif}\;a \le 1.082916304047229047679075765793884594082 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}}, \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}}, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r5337859 = x;
        double r5337860 = y;
        double r5337861 = z;
        double r5337862 = r5337860 - r5337861;
        double r5337863 = t;
        double r5337864 = r5337863 - r5337859;
        double r5337865 = a;
        double r5337866 = r5337865 - r5337861;
        double r5337867 = r5337864 / r5337866;
        double r5337868 = r5337862 * r5337867;
        double r5337869 = r5337859 + r5337868;
        return r5337869;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r5337870 = a;
        double r5337871 = -2.0167221434791827e-79;
        bool r5337872 = r5337870 <= r5337871;
        double r5337873 = x;
        double r5337874 = y;
        double r5337875 = z;
        double r5337876 = r5337874 - r5337875;
        double r5337877 = r5337870 - r5337875;
        double r5337878 = cbrt(r5337877);
        double r5337879 = r5337878 * r5337878;
        double r5337880 = r5337876 / r5337879;
        double r5337881 = t;
        double r5337882 = r5337881 - r5337873;
        double r5337883 = r5337878 / r5337882;
        double r5337884 = r5337880 / r5337883;
        double r5337885 = r5337873 + r5337884;
        double r5337886 = 1.082916304047229e-175;
        bool r5337887 = r5337870 <= r5337886;
        double r5337888 = r5337873 / r5337875;
        double r5337889 = r5337881 / r5337875;
        double r5337890 = r5337888 - r5337889;
        double r5337891 = fma(r5337874, r5337890, r5337881);
        double r5337892 = 1.0;
        double r5337893 = cbrt(r5337882);
        double r5337894 = r5337893 * r5337893;
        double r5337895 = r5337879 / r5337894;
        double r5337896 = r5337892 / r5337895;
        double r5337897 = r5337878 / r5337893;
        double r5337898 = r5337876 / r5337897;
        double r5337899 = fma(r5337896, r5337898, r5337873);
        double r5337900 = r5337887 ? r5337891 : r5337899;
        double r5337901 = r5337872 ? r5337885 : r5337900;
        return r5337901;
}

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 a < -2.0167221434791827e-79

   1. Initial program 10.3

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

   2. Simplified 10.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)}\]
   3. Using strategy rm

   4. Applied div-inv 10.2

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\left(t - x\right) \cdot \frac{1}{a - z}}, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 10.3

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

   7. Simplified 10.3

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

   9. Applied *-un-lft-identity 10.3

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

   10. Applied add-cube-cbrt 10.7

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

   11. Applied times-frac 10.7

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

   12. Applied associate-/r* 8.5

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

   13. Simplified 8.5

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

   if -2.0167221434791827e-79 < a < 1.082916304047229e-175

   1. Initial program 24.6

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

   2. Simplified 24.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)}\]

   3. Taylor expanded around inf 15.4

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

   4. Simplified 12.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)}\]

   if 1.082916304047229e-175 < a

   1. Initial program 12.1

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

   2. Simplified 12.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)}\]
   3. Using strategy rm

   4. Applied div-inv 12.1

      \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\left(t - x\right) \cdot \frac{1}{a - z}}, x\right)\]
   5. Using strategy rm

   6. Applied fma-udef 12.2

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

   7. Simplified 12.2

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

   9. Applied add-cube-cbrt 12.7

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

   10. Applied add-cube-cbrt 12.9

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

   11. Applied times-frac 12.9

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

   12. Applied *-un-lft-identity 12.9

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

   13. Applied times-frac 10.1

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

   14. Applied fma-def 10.1

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.016722143479182721107069384915645128898 \cdot 10^{-79}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\frac{\sqrt[3]{a - z}}{t - x}}\\ \mathbf{elif}\;a \le 1.082916304047229047679075765793884594082 \cdot 10^{-175}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}}, \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  (+ x (* (- y z) (/ (- t x) (- a z)))))