Average Error: 11.1 → 7.9
Time: 12.0s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\frac{x}{t - a \cdot z} - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \left(z \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t - a \cdot z}}\right)\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\frac{x}{t - a \cdot z} - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{t - a \cdot z} \cdot \sqrt[3]{t - a \cdot z}} \cdot \left(z \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{t - a \cdot z}}\right)
double f(double x, double y, double z, double t, double a) {
        double r585959 = x;
        double r585960 = y;
        double r585961 = z;
        double r585962 = r585960 * r585961;
        double r585963 = r585959 - r585962;
        double r585964 = t;
        double r585965 = a;
        double r585966 = r585965 * r585961;
        double r585967 = r585964 - r585966;
        double r585968 = r585963 / r585967;
        return r585968;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r585969 = x;
        double r585970 = t;
        double r585971 = a;
        double r585972 = z;
        double r585973 = r585971 * r585972;
        double r585974 = r585970 - r585973;
        double r585975 = r585969 / r585974;
        double r585976 = y;
        double r585977 = cbrt(r585976);
        double r585978 = r585977 * r585977;
        double r585979 = cbrt(r585974);
        double r585980 = r585979 * r585979;
        double r585981 = r585978 / r585980;
        double r585982 = r585977 / r585979;
        double r585983 = r585972 * r585982;
        double r585984 = r585981 * r585983;
        double r585985 = r585975 - r585984;
        return r585985;
}

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

Original11.1
Target1.7
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Initial program 11.1

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

  3. Applied div-sub11.1

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

  4. Simplified10.4

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

  6. Applied add-cube-cbrt10.7

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

  7. Applied add-cube-cbrt10.8

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

  8. Applied times-frac10.8

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

  9. Applied associate-*l*7.9

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

  10. Simplified7.9

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

  11. Final simplification7.9

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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