Average Error: 24.3 → 9.0
Time: 22.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\left(\frac{a}{z - t} \cdot \frac{a}{z - t}\right) \cdot \frac{a}{z - t}} - \frac{t}{z - t}}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.828093467919815950939216553514049710649 \cdot 10^{-286}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 7.220777503459318304277048324694986155104 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.802679180745016716490855755420951351133 \cdot 10^{221}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\left(\frac{a}{z - t} \cdot \frac{a}{z - t}\right) \cdot \frac{a}{z - t}} - \frac{t}{z - t}}, y - x, x\right)\\ \end{array}\]
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} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\left(\frac{a}{z - t} \cdot \frac{a}{z - t}\right) \cdot \frac{a}{z - t}} - \frac{t}{z - t}}, y - x, x\right)\\

\mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.828093467919815950939216553514049710649 \cdot 10^{-286}:\\
\;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\

\mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 7.220777503459318304277048324694986155104 \cdot 10^{-276}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\

\mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.802679180745016716490855755420951351133 \cdot 10^{221}:\\
\;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r25419439 = x;
        double r25419440 = y;
        double r25419441 = r25419440 - r25419439;
        double r25419442 = z;
        double r25419443 = t;
        double r25419444 = r25419442 - r25419443;
        double r25419445 = r25419441 * r25419444;
        double r25419446 = a;
        double r25419447 = r25419446 - r25419443;
        double r25419448 = r25419445 / r25419447;
        double r25419449 = r25419439 + r25419448;
        return r25419449;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25419450 = x;
        double r25419451 = y;
        double r25419452 = r25419451 - r25419450;
        double r25419453 = z;
        double r25419454 = t;
        double r25419455 = r25419453 - r25419454;
        double r25419456 = r25419452 * r25419455;
        double r25419457 = a;
        double r25419458 = r25419457 - r25419454;
        double r25419459 = r25419456 / r25419458;
        double r25419460 = r25419450 + r25419459;
        double r25419461 = -inf.0;
        bool r25419462 = r25419460 <= r25419461;
        double r25419463 = 1.0;
        double r25419464 = r25419457 / r25419455;
        double r25419465 = r25419464 * r25419464;
        double r25419466 = r25419465 * r25419464;
        double r25419467 = cbrt(r25419466);
        double r25419468 = r25419454 / r25419455;
        double r25419469 = r25419467 - r25419468;
        double r25419470 = r25419463 / r25419469;
        double r25419471 = fma(r25419470, r25419452, r25419450);
        double r25419472 = -1.828093467919816e-286;
        bool r25419473 = r25419460 <= r25419472;
        double r25419474 = 7.220777503459318e-276;
        bool r25419475 = r25419460 <= r25419474;
        double r25419476 = r25419450 / r25419454;
        double r25419477 = fma(r25419476, r25419453, r25419451);
        double r25419478 = r25419453 / r25419454;
        double r25419479 = r25419451 * r25419478;
        double r25419480 = r25419477 - r25419479;
        double r25419481 = 1.8026791807450167e+221;
        bool r25419482 = r25419460 <= r25419481;
        double r25419483 = r25419482 ? r25419460 : r25419471;
        double r25419484 = r25419475 ? r25419480 : r25419483;
        double r25419485 = r25419473 ? r25419460 : r25419484;
        double r25419486 = r25419462 ? r25419471 : r25419485;
        return r25419486;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.3
Target9.2
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -inf.0 or 1.8026791807450167e+221 < (+ x (/ (* (- y x) (- z t)) (- a t)))

    1. Initial program 56.0

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

    2. Simplified15.1

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

    4. Applied clear-num15.1

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

    6. Applied div-sub15.1

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

    8. Applied add-cbrt-cube32.0

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

    9. Applied add-cbrt-cube37.6

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

    10. Applied cbrt-undiv37.7

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

    11. Simplified18.3

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

    if -inf.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.828093467919816e-286 or 7.220777503459318e-276 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 1.8026791807450167e+221

    1. Initial program 1.8

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

    if -1.828093467919816e-286 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 7.220777503459318e-276

    1. Initial program 58.0

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 20.8

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

    4. Simplified23.5

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\left(\frac{a}{z - t} \cdot \frac{a}{z - t}\right) \cdot \frac{a}{z - t}} - \frac{t}{z - t}}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.828093467919815950939216553514049710649 \cdot 10^{-286}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 7.220777503459318304277048324694986155104 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y\right) - y \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.802679180745016716490855755420951351133 \cdot 10^{221}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt[3]{\left(\frac{a}{z - t} \cdot \frac{a}{z - t}\right) \cdot \frac{a}{z - t}} - \frac{t}{z - t}}, y - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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