Average Error: 10.3 → 1.8
Time: 8.0s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.31566099741130341 \cdot 10^{-17} \lor \neg \left(z \le 0.014498322822577354\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}, \frac{x}{\sqrt[3]{t - a \cdot z}}, -\frac{y}{\frac{t}{z} - \frac{a}{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.31566099741130341 \cdot 10^{-17} \lor \neg \left(z \le 0.014498322822577354\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}, \frac{x}{\sqrt[3]{t - a \cdot z}}, -\frac{y}{\frac{t}{z} - \frac{a}{1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r719911 = x;
        double r719912 = y;
        double r719913 = z;
        double r719914 = r719912 * r719913;
        double r719915 = r719911 - r719914;
        double r719916 = t;
        double r719917 = a;
        double r719918 = r719917 * r719913;
        double r719919 = r719916 - r719918;
        double r719920 = r719915 / r719919;
        return r719920;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r719921 = z;
        double r719922 = -5.3156609974113034e-17;
        bool r719923 = r719921 <= r719922;
        double r719924 = 0.014498322822577354;
        bool r719925 = r719921 <= r719924;
        double r719926 = !r719925;
        bool r719927 = r719923 || r719926;
        double r719928 = 1.0;
        double r719929 = t;
        double r719930 = a;
        double r719931 = r719930 * r719921;
        double r719932 = r719929 - r719931;
        double r719933 = cbrt(r719932);
        double r719934 = r719933 * r719933;
        double r719935 = r719928 / r719934;
        double r719936 = x;
        double r719937 = r719936 / r719933;
        double r719938 = y;
        double r719939 = r719929 / r719921;
        double r719940 = r719930 / r719928;
        double r719941 = r719939 - r719940;
        double r719942 = r719938 / r719941;
        double r719943 = -r719942;
        double r719944 = fma(r719935, r719937, r719943);
        double r719945 = r719938 * r719921;
        double r719946 = r719936 - r719945;
        double r719947 = r719932 / r719946;
        double r719948 = r719928 / r719947;
        double r719949 = r719927 ? r719944 : r719948;
        return r719949;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.3
Target1.7
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.3156609974113034e-17 or 0.014498322822577354 < z

    1. Initial program 19.8

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied div-sub19.8

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

    5. Applied associate-/l*12.3

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

    7. Applied div-sub12.3

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

    8. Simplified2.7

      \[\leadsto \frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - \color{blue}{\frac{a}{1}}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt3.0

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

    11. Applied *-un-lft-identity3.0

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

    12. Applied times-frac3.0

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

    13. Applied fma-neg3.0

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

    if -5.3156609974113034e-17 < z < 0.014498322822577354

    1. Initial program 0.1

      \[\frac{x - y \cdot z}{t - a \cdot z}\]
    2. Using strategy rm

    3. Applied clear-num0.6

      \[\leadsto \color{blue}{\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.31566099741130341 \cdot 10^{-17} \lor \neg \left(z \le 0.014498322822577354\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}}, \frac{x}{\sqrt[3]{t - a \cdot z}}, -\frac{y}{\frac{t}{z} - \frac{a}{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t - a \cdot z}{x - y \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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