Average Error: 10.9 → 7.5
Time: 10.2s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.312879063490581500293538201678987022825 \cdot 10^{-112} \lor \neg \left(z \le 1.249127713992165749732225760124090029141 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x - y \cdot z} \cdot \sqrt[3]{x - y \cdot z}}{\frac{t - a \cdot z}{\sqrt[3]{x - y \cdot z}}}\\ \end{array}\]
\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.312879063490581500293538201678987022825 \cdot 10^{-112} \lor \neg \left(z \le 1.249127713992165749732225760124090029141 \cdot 10^{-107}\right):\\
\;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{x - y \cdot z} \cdot \sqrt[3]{x - y \cdot z}}{\frac{t - a \cdot z}{\sqrt[3]{x - y \cdot z}}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r454622 = x;
        double r454623 = y;
        double r454624 = z;
        double r454625 = r454623 * r454624;
        double r454626 = r454622 - r454625;
        double r454627 = t;
        double r454628 = a;
        double r454629 = r454628 * r454624;
        double r454630 = r454627 - r454629;
        double r454631 = r454626 / r454630;
        return r454631;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r454632 = z;
        double r454633 = -1.3128790634905815e-112;
        bool r454634 = r454632 <= r454633;
        double r454635 = 1.2491277139921657e-107;
        bool r454636 = r454632 <= r454635;
        double r454637 = !r454636;
        bool r454638 = r454634 || r454637;
        double r454639 = x;
        double r454640 = t;
        double r454641 = a;
        double r454642 = r454641 * r454632;
        double r454643 = r454640 - r454642;
        double r454644 = r454639 / r454643;
        double r454645 = y;
        double r454646 = r454643 / r454632;
        double r454647 = r454645 / r454646;
        double r454648 = r454644 - r454647;
        double r454649 = r454645 * r454632;
        double r454650 = r454639 - r454649;
        double r454651 = cbrt(r454650);
        double r454652 = r454651 * r454651;
        double r454653 = r454643 / r454651;
        double r454654 = r454652 / r454653;
        double r454655 = r454638 ? r454648 : r454654;
        return r454655;
}

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

Original10.9
Target1.6
Herbie7.5
\[\begin{array}{l} \mathbf{if}\;z \lt -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \lt 3.51395223729782958298856956410892592016 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.3128790634905815e-112 or 1.2491277139921657e-107 < z

    1. Initial program 16.0

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

    3. Applied div-sub16.0

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

    5. Applied associate-/l*10.5

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

    if -1.3128790634905815e-112 < z < 1.2491277139921657e-107

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt1.2

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

    4. Applied associate-/l*1.2

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.312879063490581500293538201678987022825 \cdot 10^{-112} \lor \neg \left(z \le 1.249127713992165749732225760124090029141 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t - a \cdot z}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x - y \cdot z} \cdot \sqrt[3]{x - y \cdot z}}{\frac{t - a \cdot z}{\sqrt[3]{x - y \cdot z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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))))