Average Error: 24.7 → 10.9
Time: 25.7s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.199730766886296219043881060711901547487 \cdot 10^{-150} \lor \neg \left(a \le 1.608687110652099000373784450707986792617 \cdot 10^{-118}\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.199730766886296219043881060711901547487 \cdot 10^{-150} \lor \neg \left(a \le 1.608687110652099000373784450707986792617 \cdot 10^{-118}\right):\\
\;\;\;\;\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r386852 = x;
        double r386853 = y;
        double r386854 = r386853 - r386852;
        double r386855 = z;
        double r386856 = t;
        double r386857 = r386855 - r386856;
        double r386858 = r386854 * r386857;
        double r386859 = a;
        double r386860 = r386859 - r386856;
        double r386861 = r386858 / r386860;
        double r386862 = r386852 + r386861;
        return r386862;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r386863 = a;
        double r386864 = -2.1997307668862962e-150;
        bool r386865 = r386863 <= r386864;
        double r386866 = 1.608687110652099e-118;
        bool r386867 = r386863 <= r386866;
        double r386868 = !r386867;
        bool r386869 = r386865 || r386868;
        double r386870 = y;
        double r386871 = x;
        double r386872 = r386870 - r386871;
        double r386873 = z;
        double r386874 = t;
        double r386875 = r386873 - r386874;
        double r386876 = cbrt(r386875);
        double r386877 = r386876 * r386876;
        double r386878 = r386863 - r386874;
        double r386879 = cbrt(r386878);
        double r386880 = r386879 * r386879;
        double r386881 = r386877 / r386880;
        double r386882 = r386872 * r386881;
        double r386883 = r386876 / r386879;
        double r386884 = r386882 * r386883;
        double r386885 = r386884 + r386871;
        double r386886 = r386871 / r386874;
        double r386887 = r386873 * r386870;
        double r386888 = r386887 / r386874;
        double r386889 = r386870 - r386888;
        double r386890 = fma(r386886, r386873, r386889);
        double r386891 = r386869 ? r386885 : r386890;
        return r386891;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.7
Target9.7
Herbie10.9
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.1997307668862962e-150 or 1.608687110652099e-118 < a

    1. Initial program 23.1

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

    2. Simplified11.6

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

    4. Applied fma-udef11.6

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

    6. Applied div-inv11.7

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

    7. Applied associate-*l*9.4

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

    8. Simplified9.4

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

    10. Applied add-cube-cbrt9.9

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

    11. Applied add-cube-cbrt9.9

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

    12. Applied times-frac9.9

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

    13. Applied associate-*r*9.5

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

    if -2.1997307668862962e-150 < a < 1.608687110652099e-118

    1. Initial program 29.4

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

    2. Simplified24.8

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

    4. Applied fma-udef24.9

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

    6. Applied div-inv24.9

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

    7. Applied associate-*l*19.6

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

    8. Simplified19.5

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

    9. Taylor expanded around inf 15.1

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

    10. Simplified15.1

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.199730766886296219043881060711901547487 \cdot 10^{-150} \lor \neg \left(a \le 1.608687110652099000373784450707986792617 \cdot 10^{-118}\right):\\ \;\;\;\;\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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