Average Error: 24.3 → 9.2
Time: 5.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.29505595240201798 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.29505595240201798 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\
\;\;\;\;\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y - x}}{\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 r632765 = x;
        double r632766 = y;
        double r632767 = r632766 - r632765;
        double r632768 = z;
        double r632769 = t;
        double r632770 = r632768 - r632769;
        double r632771 = r632767 * r632770;
        double r632772 = a;
        double r632773 = r632772 - r632769;
        double r632774 = r632771 / r632773;
        double r632775 = r632765 + r632774;
        return r632775;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r632776 = x;
        double r632777 = y;
        double r632778 = r632777 - r632776;
        double r632779 = z;
        double r632780 = t;
        double r632781 = r632779 - r632780;
        double r632782 = r632778 * r632781;
        double r632783 = a;
        double r632784 = r632783 - r632780;
        double r632785 = r632782 / r632784;
        double r632786 = r632776 + r632785;
        double r632787 = -3.295055952402018e-290;
        bool r632788 = r632786 <= r632787;
        double r632789 = 0.0;
        bool r632790 = r632786 <= r632789;
        double r632791 = !r632790;
        bool r632792 = r632788 || r632791;
        double r632793 = cbrt(r632778);
        double r632794 = r632793 * r632793;
        double r632795 = cbrt(r632784);
        double r632796 = r632795 * r632795;
        double r632797 = r632794 / r632796;
        double r632798 = r632781 * r632797;
        double r632799 = r632793 / r632795;
        double r632800 = r632798 * r632799;
        double r632801 = r632800 + r632776;
        double r632802 = r632776 / r632780;
        double r632803 = r632779 * r632777;
        double r632804 = r632803 / r632780;
        double r632805 = r632777 - r632804;
        double r632806 = fma(r632802, r632779, r632805);
        double r632807 = r632792 ? r632801 : r632806;
        return r632807;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.3
Target9.4
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -3.295055952402018e-290 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 21.2

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

    2. Simplified10.8

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

    4. Applied div-inv10.9

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

    6. Applied fma-udef10.9

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

    7. Simplified10.8

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

    9. Applied add-cube-cbrt11.4

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

    10. Applied add-cube-cbrt11.6

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

    11. Applied times-frac11.6

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

    12. Applied associate-*r*8.0

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

    if -3.295055952402018e-290 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

    1. Initial program 59.7

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

    2. Simplified59.6

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

    4. Applied div-inv59.5

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

    6. Applied fma-udef59.6

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

    7. Simplified59.8

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

    8. Taylor expanded around inf 19.3

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

    9. Simplified22.6

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -3.29505595240201798 \cdot 10^{-290} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))