Average Error: 16.6 → 8.7
Time: 16.7s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.758247285237226479214401882402671748034 \cdot 10^{-87}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \mathbf{elif}\;a \le 1.901187471214967816761071687200623055822 \cdot 10^{-101}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.758247285237226479214401882402671748034 \cdot 10^{-87}:\\
\;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\

\mathbf{elif}\;a \le 1.901187471214967816761071687200623055822 \cdot 10^{-101}:\\
\;\;\;\;\frac{z \cdot y}{t} + x\\

\mathbf{else}:\\
\;\;\;\;\left(x + y\right) - \left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r492213 = x;
        double r492214 = y;
        double r492215 = r492213 + r492214;
        double r492216 = z;
        double r492217 = t;
        double r492218 = r492216 - r492217;
        double r492219 = r492218 * r492214;
        double r492220 = a;
        double r492221 = r492220 - r492217;
        double r492222 = r492219 / r492221;
        double r492223 = r492215 - r492222;
        return r492223;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r492224 = a;
        double r492225 = -1.7582472852372265e-87;
        bool r492226 = r492224 <= r492225;
        double r492227 = x;
        double r492228 = y;
        double r492229 = r492227 + r492228;
        double r492230 = z;
        double r492231 = t;
        double r492232 = r492230 - r492231;
        double r492233 = r492224 - r492231;
        double r492234 = cbrt(r492233);
        double r492235 = r492234 * r492234;
        double r492236 = r492232 / r492235;
        double r492237 = cbrt(r492236);
        double r492238 = r492237 * r492237;
        double r492239 = cbrt(r492228);
        double r492240 = r492239 * r492239;
        double r492241 = 1.0;
        double r492242 = cbrt(r492241);
        double r492243 = r492240 / r492242;
        double r492244 = r492237 * r492243;
        double r492245 = r492238 * r492244;
        double r492246 = r492239 / r492234;
        double r492247 = r492245 * r492246;
        double r492248 = r492229 - r492247;
        double r492249 = 1.9011874712149678e-101;
        bool r492250 = r492224 <= r492249;
        double r492251 = r492230 * r492228;
        double r492252 = r492251 / r492231;
        double r492253 = r492252 + r492227;
        double r492254 = cbrt(r492232);
        double r492255 = r492254 * r492254;
        double r492256 = r492255 / r492234;
        double r492257 = r492254 / r492234;
        double r492258 = r492257 * r492243;
        double r492259 = r492256 * r492258;
        double r492260 = r492259 * r492246;
        double r492261 = r492229 - r492260;
        double r492262 = r492250 ? r492253 : r492261;
        double r492263 = r492226 ? r492248 : r492262;
        return r492263;
}

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

Original16.6
Target8.6
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.7582472852372265e-87

    1. Initial program 15.0

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt15.1

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

    4. Applied times-frac7.9

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity7.9

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

    7. Applied cbrt-prod7.9

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

    8. Applied add-cube-cbrt7.9

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

    9. Applied times-frac7.9

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

    10. Applied associate-*r*7.8

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt7.8

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

    13. Applied associate-*l*7.8

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

    if -1.7582472852372265e-87 < a < 1.9011874712149678e-101

    1. Initial program 19.9

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

    2. Taylor expanded around inf 10.7

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

    if 1.9011874712149678e-101 < a

    1. Initial program 15.0

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt15.1

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

    4. Applied times-frac8.1

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity8.1

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

    7. Applied cbrt-prod8.1

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

    8. Applied add-cube-cbrt8.2

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

    9. Applied times-frac8.1

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

    10. Applied associate-*r*8.0

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt8.0

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

    13. Applied times-frac8.0

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

    14. Applied associate-*l*7.7

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.758247285237226479214401882402671748034 \cdot 10^{-87}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \mathbf{elif}\;a \le 1.901187471214967816761071687200623055822 \cdot 10^{-101}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{1}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-7) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.47542934445772333e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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