Average Error: 16.4 → 8.0
Time: 5.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.01778378545867122 \cdot 10^{90} \lor \neg \left(t \le 6.40427683306346665 \cdot 10^{118}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\frac{t - z}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y}}}}, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -5.01778378545867122 \cdot 10^{90} \lor \neg \left(t \le 6.40427683306346665 \cdot 10^{118}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r572262 = x;
        double r572263 = y;
        double r572264 = r572262 + r572263;
        double r572265 = z;
        double r572266 = t;
        double r572267 = r572265 - r572266;
        double r572268 = r572267 * r572263;
        double r572269 = a;
        double r572270 = r572269 - r572266;
        double r572271 = r572268 / r572270;
        double r572272 = r572264 - r572271;
        return r572272;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r572273 = t;
        double r572274 = -5.017783785458671e+90;
        bool r572275 = r572273 <= r572274;
        double r572276 = 6.404276833063467e+118;
        bool r572277 = r572273 <= r572276;
        double r572278 = !r572277;
        bool r572279 = r572275 || r572278;
        double r572280 = z;
        double r572281 = r572280 / r572273;
        double r572282 = y;
        double r572283 = x;
        double r572284 = fma(r572281, r572282, r572283);
        double r572285 = 1.0;
        double r572286 = cbrt(r572282);
        double r572287 = r572286 * r572286;
        double r572288 = r572285 / r572287;
        double r572289 = r572285 / r572288;
        double r572290 = r572273 - r572280;
        double r572291 = a;
        double r572292 = r572291 - r572273;
        double r572293 = cbrt(r572292);
        double r572294 = r572293 * r572293;
        double r572295 = cbrt(r572286);
        double r572296 = r572295 * r572295;
        double r572297 = r572294 / r572296;
        double r572298 = r572290 / r572297;
        double r572299 = r572293 / r572295;
        double r572300 = r572298 / r572299;
        double r572301 = r572283 + r572282;
        double r572302 = fma(r572289, r572300, r572301);
        double r572303 = r572279 ? r572284 : r572302;
        return r572303;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.4
Target8.5
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -5.017783785458671e+90 or 6.404276833063467e+118 < t

    1. Initial program 29.8

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

    2. Simplified20.7

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

    4. Applied clear-num20.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
    5. Using strategy rm

    6. Applied fma-udef20.9

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

    7. Simplified20.9

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

    8. Taylor expanded around inf 17.3

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

    9. Simplified12.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}\]

    if -5.017783785458671e+90 < t < 6.404276833063467e+118

    1. Initial program 9.0

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

    2. Simplified6.3

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

    4. Applied clear-num6.4

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
    5. Using strategy rm

    6. Applied fma-udef6.4

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

    7. Simplified6.2

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

    9. Applied add-cube-cbrt6.4

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

    10. Applied *-un-lft-identity6.4

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

    11. Applied times-frac6.4

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

    12. Applied *-un-lft-identity6.4

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

    13. Applied times-frac5.8

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

    14. Applied fma-def5.7

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

    16. Applied add-cube-cbrt5.8

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

    17. Applied add-cube-cbrt5.9

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

    18. Applied times-frac5.9

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

    19. Applied associate-/r*5.8

      \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \color{blue}{\frac{\frac{t - z}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y}}}}}, x + y\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.01778378545867122 \cdot 10^{90} \lor \neg \left(t \le 6.40427683306346665 \cdot 10^{118}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\frac{t - z}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{y}}}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))