Average Error: 16.1 → 9.6
Time: 26.4s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.91570009620441830670906157024910521968 \cdot 10^{107} \lor \neg \left(t \le 3.351252885723445614738514992744822267851 \cdot 10^{128}\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \left(\left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right) \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.91570009620441830670906157024910521968 \cdot 10^{107} \lor \neg \left(t \le 3.351252885723445614738514992744822267851 \cdot 10^{128}\right):\\
\;\;\;\;\frac{z \cdot y}{t} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r411929 = x;
        double r411930 = y;
        double r411931 = r411929 + r411930;
        double r411932 = z;
        double r411933 = t;
        double r411934 = r411932 - r411933;
        double r411935 = r411934 * r411930;
        double r411936 = a;
        double r411937 = r411936 - r411933;
        double r411938 = r411935 / r411937;
        double r411939 = r411931 - r411938;
        return r411939;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r411940 = t;
        double r411941 = -1.9157000962044183e+107;
        bool r411942 = r411940 <= r411941;
        double r411943 = 3.3512528857234456e+128;
        bool r411944 = r411940 <= r411943;
        double r411945 = !r411944;
        bool r411946 = r411942 || r411945;
        double r411947 = z;
        double r411948 = y;
        double r411949 = r411947 * r411948;
        double r411950 = r411949 / r411940;
        double r411951 = x;
        double r411952 = r411950 + r411951;
        double r411953 = r411951 + r411948;
        double r411954 = r411947 - r411940;
        double r411955 = cbrt(r411954);
        double r411956 = cbrt(r411955);
        double r411957 = r411956 * r411956;
        double r411958 = r411957 * r411956;
        double r411959 = r411955 * r411958;
        double r411960 = a;
        double r411961 = r411960 - r411940;
        double r411962 = cbrt(r411961);
        double r411963 = r411959 / r411962;
        double r411964 = r411955 / r411962;
        double r411965 = r411948 / r411962;
        double r411966 = r411964 * r411965;
        double r411967 = r411963 * r411966;
        double r411968 = r411953 - r411967;
        double r411969 = r411946 ? r411952 : r411968;
        return r411969;
}

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

Original16.1
Target8.1
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.9157000962044183e+107 or 3.3512528857234456e+128 < t

    1. Initial program 29.7

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

    2. Taylor expanded around inf 17.3

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

    if -1.9157000962044183e+107 < t < 3.3512528857234456e+128

    1. Initial program 9.1

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

    3. Applied add-cube-cbrt9.3

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

    4. Applied times-frac6.4

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

    6. Applied add-cube-cbrt6.4

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

    7. Applied times-frac6.4

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

    8. Applied associate-*l*5.6

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

    10. Applied add-cube-cbrt5.6

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.91570009620441830670906157024910521968 \cdot 10^{107} \lor \neg \left(t \le 3.351252885723445614738514992744822267851 \cdot 10^{128}\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \left(\left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right) \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))))