Average Error: 24.4 → 9.9
Time: 16.7s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -3.21462567963864648214684556864782241168 \cdot 10^{-139} \lor \neg \left(a \le 9.616031131563334674698731158316533821258 \cdot 10^{-161}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{1}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]
x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -3.21462567963864648214684556864782241168 \cdot 10^{-139} \lor \neg \left(a \le 9.616031131563334674698731158316533821258 \cdot 10^{-161}\right):\\
\;\;\;\;x + \left(y - x\right) \cdot \frac{1}{\frac{a - t}{z - t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r530457 = x;
        double r530458 = y;
        double r530459 = r530458 - r530457;
        double r530460 = z;
        double r530461 = t;
        double r530462 = r530460 - r530461;
        double r530463 = r530459 * r530462;
        double r530464 = a;
        double r530465 = r530464 - r530461;
        double r530466 = r530463 / r530465;
        double r530467 = r530457 + r530466;
        return r530467;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r530468 = a;
        double r530469 = -3.2146256796386465e-139;
        bool r530470 = r530468 <= r530469;
        double r530471 = 9.616031131563335e-161;
        bool r530472 = r530468 <= r530471;
        double r530473 = !r530472;
        bool r530474 = r530470 || r530473;
        double r530475 = x;
        double r530476 = y;
        double r530477 = r530476 - r530475;
        double r530478 = 1.0;
        double r530479 = t;
        double r530480 = r530468 - r530479;
        double r530481 = z;
        double r530482 = r530481 - r530479;
        double r530483 = r530480 / r530482;
        double r530484 = r530478 / r530483;
        double r530485 = r530477 * r530484;
        double r530486 = r530475 + r530485;
        double r530487 = r530475 * r530481;
        double r530488 = r530487 / r530479;
        double r530489 = r530476 + r530488;
        double r530490 = r530481 * r530476;
        double r530491 = r530490 / r530479;
        double r530492 = r530489 - r530491;
        double r530493 = r530474 ? r530486 : r530492;
        return r530493;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original24.4
Target9.1
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -3.2146256796386465e-139 or 9.616031131563335e-161 < a

    1. Initial program 22.9

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

    3. Applied associate-/l*9.2

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{a - t}{z - t}}}\]
    4. Using strategy rm

    5. Applied pow19.2

      \[\leadsto x + \frac{y - x}{\color{blue}{{\left(\frac{a - t}{z - t}\right)}^{1}}}\]
    6. Using strategy rm

    7. Applied div-inv9.2

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

    8. Simplified9.2

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

    if -3.2146256796386465e-139 < a < 9.616031131563335e-161

    1. Initial program 29.6

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

    2. Taylor expanded around inf 12.4

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.21462567963864648214684556864782241168 \cdot 10^{-139} \lor \neg \left(a \le 9.616031131563334674698731158316533821258 \cdot 10^{-161}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{1}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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 3.7744031700831742e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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