Average Error: 10.9 → 2.1
Time: 8.8s
Precision: 64
\[\frac{x - y \cdot z}{t - a \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.312879063490581500293538201678987022825 \cdot 10^{-112} \lor \neg \left(z \le 1.249127713992165749732225760124090029141 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{-y}{-\left(\frac{t}{z} - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x - y \cdot z} \cdot \sqrt[3]{x - y \cdot z}}{\frac{t - a \cdot z}{\sqrt[3]{x - y \cdot z}}}\\ \end{array}\]
\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}{-\left(\frac{t}{z} - a\right)}\\

\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 r514382 = x;
        double r514383 = y;
        double r514384 = z;
        double r514385 = r514383 * r514384;
        double r514386 = r514382 - r514385;
        double r514387 = t;
        double r514388 = a;
        double r514389 = r514388 * r514384;
        double r514390 = r514387 - r514389;
        double r514391 = r514386 / r514390;
        return r514391;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r514392 = z;
        double r514393 = -1.3128790634905815e-112;
        bool r514394 = r514392 <= r514393;
        double r514395 = 1.2491277139921657e-107;
        bool r514396 = r514392 <= r514395;
        double r514397 = !r514396;
        bool r514398 = r514394 || r514397;
        double r514399 = x;
        double r514400 = t;
        double r514401 = a;
        double r514402 = r514401 * r514392;
        double r514403 = r514400 - r514402;
        double r514404 = r514399 / r514403;
        double r514405 = y;
        double r514406 = -r514405;
        double r514407 = r514400 / r514392;
        double r514408 = r514407 - r514401;
        double r514409 = -r514408;
        double r514410 = r514406 / r514409;
        double r514411 = r514404 - r514410;
        double r514412 = r514405 * r514392;
        double r514413 = r514399 - r514412;
        double r514414 = cbrt(r514413);
        double r514415 = r514414 * r514414;
        double r514416 = r514403 / r514414;
        double r514417 = r514415 / r514416;
        double r514418 = r514398 ? r514411 : r514417;
        return r514418;
}

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

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

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

    7. Applied frac-2neg10.5

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

    8. Simplified2.5

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

    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 simplification2.1

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

Reproduce

herbie shell --seed 2019305 
(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))))