Average Error: 24.6 → 9.9
Time: 28.8s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.72909184718930696180304451348657521234 \cdot 10^{115} \lor \neg \left(z \le 5.707371382778094428202635106510771226033 \cdot 10^{189}\right):\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot \mathsf{fma}\left(1, t, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(x, -1, x\right), \frac{y - z}{a - z}, x\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.72909184718930696180304451348657521234 \cdot 10^{115} \lor \neg \left(z \le 5.707371382778094428202635106510771226033 \cdot 10^{189}\right):\\
\;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r370407 = x;
        double r370408 = y;
        double r370409 = z;
        double r370410 = r370408 - r370409;
        double r370411 = t;
        double r370412 = r370411 - r370407;
        double r370413 = r370410 * r370412;
        double r370414 = a;
        double r370415 = r370414 - r370409;
        double r370416 = r370413 / r370415;
        double r370417 = r370407 + r370416;
        return r370417;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r370418 = z;
        double r370419 = -2.729091847189307e+115;
        bool r370420 = r370418 <= r370419;
        double r370421 = 5.707371382778094e+189;
        bool r370422 = r370418 <= r370421;
        double r370423 = !r370422;
        bool r370424 = r370420 || r370423;
        double r370425 = t;
        double r370426 = y;
        double r370427 = r370426 / r370418;
        double r370428 = x;
        double r370429 = r370425 - r370428;
        double r370430 = r370427 * r370429;
        double r370431 = r370425 - r370430;
        double r370432 = r370426 - r370418;
        double r370433 = a;
        double r370434 = r370433 - r370418;
        double r370435 = r370432 / r370434;
        double r370436 = 1.0;
        double r370437 = cbrt(r370428);
        double r370438 = r370437 * r370437;
        double r370439 = r370437 * r370438;
        double r370440 = -r370439;
        double r370441 = fma(r370436, r370425, r370440);
        double r370442 = r370435 * r370441;
        double r370443 = -1.0;
        double r370444 = fma(r370428, r370443, r370428);
        double r370445 = fma(r370444, r370435, r370428);
        double r370446 = r370442 + r370445;
        double r370447 = r370424 ? r370431 : r370446;
        return r370447;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.6
Target11.9
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;z \lt -1.253613105609503593846459977496550767343 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.729091847189307e+115 or 5.707371382778094e+189 < z

    1. Initial program 47.6

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

    2. Simplified23.2

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

    3. Taylor expanded around inf 25.4

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

    4. Simplified15.1

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

    if -2.729091847189307e+115 < z < 5.707371382778094e+189

    1. Initial program 15.4

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

    2. Simplified7.6

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

    4. Applied fma-udef7.6

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

    6. Applied add-cube-cbrt7.8

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

    7. Applied *-un-lft-identity7.8

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

    8. Applied prod-diff7.8

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

    9. Applied distribute-lft-in7.8

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

    10. Applied associate-+l+7.7

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

    11. Simplified7.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.72909184718930696180304451348657521234 \cdot 10^{115} \lor \neg \left(z \le 5.707371382778094428202635106510771226033 \cdot 10^{189}\right):\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot \mathsf{fma}\left(1, t, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(x, -1, x\right), \frac{y - z}{a - z}, x\right)\\ \end{array}\]

Reproduce

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

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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