Average Error: 16.5 → 8.2
Time: 12.3s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.22266 \cdot 10^{-261}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{3}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.246342234530279 \cdot 10^{-220}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.22266 \cdot 10^{-261}:\\
\;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{3}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\

\mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.246342234530279 \cdot 10^{-220}:\\
\;\;\;\;\frac{z \cdot y}{t} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r685080 = x;
        double r685081 = y;
        double r685082 = r685080 + r685081;
        double r685083 = z;
        double r685084 = t;
        double r685085 = r685083 - r685084;
        double r685086 = r685085 * r685081;
        double r685087 = a;
        double r685088 = r685087 - r685084;
        double r685089 = r685086 / r685088;
        double r685090 = r685082 - r685089;
        return r685090;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r685091 = x;
        double r685092 = y;
        double r685093 = r685091 + r685092;
        double r685094 = z;
        double r685095 = t;
        double r685096 = r685094 - r685095;
        double r685097 = r685096 * r685092;
        double r685098 = a;
        double r685099 = r685098 - r685095;
        double r685100 = r685097 / r685099;
        double r685101 = r685093 - r685100;
        double r685102 = -1.222656639302079e-261;
        bool r685103 = r685101 <= r685102;
        double r685104 = cbrt(r685099);
        double r685105 = r685104 * r685104;
        double r685106 = r685096 / r685105;
        double r685107 = cbrt(r685092);
        double r685108 = r685107 * r685107;
        double r685109 = cbrt(r685105);
        double r685110 = r685108 / r685109;
        double r685111 = r685106 * r685110;
        double r685112 = 3.0;
        double r685113 = pow(r685107, r685112);
        double r685114 = cbrt(r685113);
        double r685115 = cbrt(r685104);
        double r685116 = r685114 / r685115;
        double r685117 = r685111 * r685116;
        double r685118 = r685093 - r685117;
        double r685119 = 2.2463422345302793e-220;
        bool r685120 = r685101 <= r685119;
        double r685121 = r685094 * r685092;
        double r685122 = r685121 / r685095;
        double r685123 = r685122 + r685091;
        double r685124 = r685092 / r685104;
        double r685125 = r685106 * r685124;
        double r685126 = r685093 - r685125;
        double r685127 = r685120 ? r685123 : r685126;
        double r685128 = r685103 ? r685118 : r685127;
        return r685128;
}

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

Original16.5
Target8.5
Herbie8.2
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -1.222656639302079e-261

    1. Initial program 12.9

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied times-frac7.5

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

    6. Applied add-cube-cbrt7.6

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

    7. Applied cbrt-prod7.6

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

    8. Applied add-cube-cbrt7.6

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

    9. Applied times-frac7.6

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

    10. Applied associate-*r*7.3

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

    12. Applied add-cbrt-cube7.3

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

    13. Simplified7.3

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

    if -1.222656639302079e-261 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 2.2463422345302793e-220

    1. Initial program 54.5

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

    2. Taylor expanded around inf 16.9

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

    if 2.2463422345302793e-220 < (- (+ x y) (/ (* (- z t) y) (- a t)))

    1. Initial program 12.9

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied times-frac7.5

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.22266 \cdot 10^{-261}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt[3]{y}\right)}^{3}}}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 2.246342234530279 \cdot 10^{-220}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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