Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Average Error: 1.9 → 2.0

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.40921362095974592 \cdot 10^{116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-1 \cdot z}{-y}, x, -\frac{x}{y} \cdot t\right) + \frac{x}{y} \cdot \mathsf{fma}\left(-t, 1, t\right)\right) + t\\ \mathbf{elif}\;y \le 2.38756589134765261 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r609927 = x;
        double r609928 = y;
        double r609929 = r609927 / r609928;
        double r609930 = z;
        double r609931 = t;
        double r609932 = r609930 - r609931;
        double r609933 = r609929 * r609932;
        double r609934 = r609933 + r609931;
        return r609934;
}

↓

double f(double x, double y, double z, double t) {
        double r609935 = y;
        double r609936 = -4.409213620959746e+116;
        bool r609937 = r609935 <= r609936;
        double r609938 = -1.0;
        double r609939 = z;
        double r609940 = r609938 * r609939;
        double r609941 = -r609935;
        double r609942 = r609940 / r609941;
        double r609943 = x;
        double r609944 = r609943 / r609935;
        double r609945 = t;
        double r609946 = r609944 * r609945;
        double r609947 = -r609946;
        double r609948 = fma(r609942, r609943, r609947);
        double r609949 = -r609945;
        double r609950 = 1.0;
        double r609951 = fma(r609949, r609950, r609945);
        double r609952 = r609944 * r609951;
        double r609953 = r609948 + r609952;
        double r609954 = r609953 + r609945;
        double r609955 = 2.3875658913476526e-163;
        bool r609956 = r609935 <= r609955;
        double r609957 = r609939 - r609945;
        double r609958 = r609943 * r609957;
        double r609959 = r609958 / r609935;
        double r609960 = r609959 + r609945;
        double r609961 = r609944 * r609957;
        double r609962 = r609961 + r609945;
        double r609963 = r609956 ? r609960 : r609962;
        double r609964 = r609937 ? r609954 : r609963;
        return r609964;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.9
Target	2.3
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -4.409213620959746e+116

   1. Initial program 1.1

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

   3. Applied add-cube-cbrt 1.2

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

   4. Applied add-cube-cbrt 1.4

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

   5. Applied prod-diff 1.4

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

   6. Applied distribute-lft-in 1.4

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

   7. Simplified 1.4

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

   8. Simplified 1.4

      \[\leadsto \left(\left(\frac{{\left(\sqrt[3]{z}\right)}^{3}}{\frac{y}{x}} - \frac{x}{y} \cdot t\right) + \color{blue}{\frac{x}{y} \cdot \mathsf{fma}\left(-t, 1, t\right)}\right) + t\]
   9. Using strategy rm

   10. Applied frac-2neg 1.4

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

   11. Simplified 1.2

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

   13. Applied distribute-neg-frac 1.2

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

   14. Applied associate-/r/ 1.0

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

   15. Applied fma-neg 1.0

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

   if -4.409213620959746e+116 < y < 2.3875658913476526e-163

   1. Initial program 3.1

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

   3. Applied associate-*l/ 3.4

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

   if 2.3875658913476526e-163 < y

   1. Initial program 1.3

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.40921362095974592 \cdot 10^{116}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{-1 \cdot z}{-y}, x, -\frac{x}{y} \cdot t\right) + \frac{x}{y} \cdot \mathsf{fma}\left(-t, 1, t\right)\right) + t\\ \mathbf{elif}\;y \le 2.38756589134765261 \cdot 10^{-163}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

  (+ (* (/ x y) (- z t)) t))