Average Error: 24.5 → 10.1
Time: 17.6s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.10252611137827803 \cdot 10^{112} \lor \neg \left(z \le 1.30984479346046122 \cdot 10^{176}\right):\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.10252611137827803 \cdot 10^{112} \lor \neg \left(z \le 1.30984479346046122 \cdot 10^{176}\right):\\
\;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r559415 = x;
        double r559416 = y;
        double r559417 = z;
        double r559418 = r559416 - r559417;
        double r559419 = t;
        double r559420 = r559419 - r559415;
        double r559421 = r559418 * r559420;
        double r559422 = a;
        double r559423 = r559422 - r559417;
        double r559424 = r559421 / r559423;
        double r559425 = r559415 + r559424;
        return r559425;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r559426 = z;
        double r559427 = -3.102526111378278e+112;
        bool r559428 = r559426 <= r559427;
        double r559429 = 1.3098447934604612e+176;
        bool r559430 = r559426 <= r559429;
        double r559431 = !r559430;
        bool r559432 = r559428 || r559431;
        double r559433 = y;
        double r559434 = x;
        double r559435 = r559434 / r559426;
        double r559436 = t;
        double r559437 = r559436 / r559426;
        double r559438 = r559435 - r559437;
        double r559439 = r559433 * r559438;
        double r559440 = r559439 + r559436;
        double r559441 = r559433 - r559426;
        double r559442 = a;
        double r559443 = r559442 - r559426;
        double r559444 = r559441 / r559443;
        double r559445 = r559436 - r559434;
        double r559446 = fma(r559444, r559445, r559434);
        double r559447 = r559432 ? r559440 : r559446;
        return r559447;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original24.5
Target11.7
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.102526111378278e+112 or 1.3098447934604612e+176 < z

    1. Initial program 46.6

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

    2. Simplified22.1

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

    4. Applied div-inv22.2

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

    5. Taylor expanded around inf 26.2

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

    6. Simplified16.8

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

    if -3.102526111378278e+112 < z < 1.3098447934604612e+176

    1. Initial program 15.2

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

    2. Simplified7.2

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

    4. Applied div-inv7.3

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

    6. Applied un-div-inv7.2

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.10252611137827803 \cdot 10^{112} \lor \neg \left(z \le 1.30984479346046122 \cdot 10^{176}\right):\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))