Average Error: 10.7 → 2.5
Time: 18.6s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -60014490737444068547486232894595568509120000:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 1.889597509550593858319042345597883847546 \cdot 10^{105}:\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\right) + \frac{y}{\frac{t}{z} - a} \cdot 0\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -60014490737444068547486232894595568509120000:\\
\;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\

\mathbf{elif}\;z \le 1.889597509550593858319042345597883847546 \cdot 10^{105}:\\
\;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\left(\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} - a}\right) + \frac{y}{\frac{t}{z} - a} \cdot 0\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r1183910 = x;
        double r1183911 = y;
        double r1183912 = z;
        double r1183913 = r1183911 * r1183912;
        double r1183914 = r1183910 - r1183913;
        double r1183915 = t;
        double r1183916 = a;
        double r1183917 = r1183916 * r1183912;
        double r1183918 = r1183915 - r1183917;
        double r1183919 = r1183914 / r1183918;
        return r1183919;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1183920 = z;
        double r1183921 = -6.001449073744407e+43;
        bool r1183922 = r1183920 <= r1183921;
        double r1183923 = x;
        double r1183924 = t;
        double r1183925 = a;
        double r1183926 = r1183925 * r1183920;
        double r1183927 = r1183924 - r1183926;
        double r1183928 = r1183923 / r1183927;
        double r1183929 = cbrt(r1183928);
        double r1183930 = r1183929 * r1183929;
        double r1183931 = r1183930 * r1183929;
        double r1183932 = y;
        double r1183933 = r1183924 / r1183920;
        double r1183934 = r1183933 - r1183925;
        double r1183935 = r1183932 / r1183934;
        double r1183936 = r1183931 - r1183935;
        double r1183937 = 1.8895975095505939e+105;
        bool r1183938 = r1183920 <= r1183937;
        double r1183939 = 1.0;
        double r1183940 = r1183939 / r1183927;
        double r1183941 = r1183923 * r1183940;
        double r1183942 = r1183932 * r1183920;
        double r1183943 = r1183942 / r1183927;
        double r1183944 = r1183941 - r1183943;
        double r1183945 = cbrt(r1183927);
        double r1183946 = r1183945 * r1183945;
        double r1183947 = r1183939 / r1183946;
        double r1183948 = r1183923 / r1183945;
        double r1183949 = r1183947 * r1183948;
        double r1183950 = r1183949 - r1183935;
        double r1183951 = 0.0;
        double r1183952 = r1183935 * r1183951;
        double r1183953 = r1183950 + r1183952;
        double r1183954 = r1183938 ? r1183944 : r1183953;
        double r1183955 = r1183922 ? r1183936 : r1183954;
        return r1183955;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.7
Target1.8
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -6.001449073744407e+43

    1. Initial program 23.7

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

    3. Applied div-sub23.7

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

    5. Applied associate-/l*13.5

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

    6. Taylor expanded around 0 3.6

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

    8. Applied add-cube-cbrt3.8

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

    if -6.001449073744407e+43 < z < 1.8895975095505939e+105

    1. Initial program 1.5

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

    3. Applied div-sub1.5

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

    5. Applied div-inv1.6

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

    if 1.8895975095505939e+105 < z

    1. Initial program 28.5

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

    3. Applied div-sub28.5

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

    5. Applied associate-/l*18.4

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

    6. Taylor expanded around 0 3.8

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

    8. Applied add-cube-cbrt4.7

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

    9. Applied add-cube-cbrt4.9

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

    10. Applied *-un-lft-identity4.9

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

    11. Applied times-frac4.9

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

    12. Applied prod-diff4.9

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

    13. Simplified4.0

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

    14. Simplified4.0

      \[\leadsto \left(\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\right) + \color{blue}{\frac{y}{\frac{t}{z} - a} \cdot 0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -60014490737444068547486232894595568509120000:\\ \;\;\;\;\left(\sqrt[3]{\frac{x}{t - a \cdot z}} \cdot \sqrt[3]{\frac{x}{t - a \cdot z}}\right) \cdot \sqrt[3]{\frac{x}{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \le 1.889597509550593858319042345597883847546 \cdot 10^{105}:\\ \;\;\;\;x \cdot \frac{1}{t - a \cdot z} - \frac{y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \frac{x}{\sqrt[3]{t - a \cdot z}} - \frac{y}{\frac{t}{z} - a}\right) + \frac{y}{\frac{t}{z} - a} \cdot 0\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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 3.51395223729782958e-86) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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