Average Error: 4.5 → 4.3
Time: 13.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.740368709049869308088666682154706200096 \cdot 10^{-97} \lor \neg \left(t \le 2.823511479814487689030883192213886222863 \cdot 10^{-176}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.740368709049869308088666682154706200096 \cdot 10^{-97} \lor \neg \left(t \le 2.823511479814487689030883192213886222863 \cdot 10^{-176}\right):\\
\;\;\;\;\frac{x}{\frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{\frac{z}{x}} + \left(-\frac{t \cdot x}{1 - z}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r277327 = x;
        double r277328 = y;
        double r277329 = z;
        double r277330 = r277328 / r277329;
        double r277331 = t;
        double r277332 = 1.0;
        double r277333 = r277332 - r277329;
        double r277334 = r277331 / r277333;
        double r277335 = r277330 - r277334;
        double r277336 = r277327 * r277335;
        return r277336;
}
↓
double f(double x, double y, double z, double t) {
        double r277337 = t;
        double r277338 = -2.7403687090498693e-97;
        bool r277339 = r277337 <= r277338;
        double r277340 = 2.8235114798144877e-176;
        bool r277341 = r277337 <= r277340;
        double r277342 = !r277341;
        bool r277343 = r277339 || r277342;
        double r277344 = x;
        double r277345 = z;
        double r277346 = y;
        double r277347 = r277345 / r277346;
        double r277348 = r277344 / r277347;
        double r277349 = 1.0;
        double r277350 = r277349 - r277345;
        double r277351 = r277337 / r277350;
        double r277352 = -r277351;
        double r277353 = r277352 * r277344;
        double r277354 = r277348 + r277353;
        double r277355 = r277345 / r277344;
        double r277356 = r277346 / r277355;
        double r277357 = r277337 * r277344;
        double r277358 = r277357 / r277350;
        double r277359 = -r277358;
        double r277360 = r277356 + r277359;
        double r277361 = r277343 ? r277354 : r277360;
        return r277361;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.5
Target4.2
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -2.7403687090498693e-97 or 2.8235114798144877e-176 < t

    1. Initial program 4.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg4.0

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

    4. Applied distribute-lft-in4.0

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

    5. Simplified4.7

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

    6. Simplified4.7

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

    8. Applied associate-/l*3.7

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

    if -2.7403687090498693e-97 < t < 2.8235114798144877e-176

    1. Initial program 5.8

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied sub-neg5.8

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

    4. Applied distribute-lft-in5.8

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

    5. Simplified7.4

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

    6. Simplified7.4

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

    7. Taylor expanded around 0 7.4

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

    8. Simplified6.9

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

    10. Applied *-un-lft-identity6.9

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

    11. Applied *-un-lft-identity6.9

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

    12. Applied times-frac6.9

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

    13. Applied distribute-lft-neg-in6.9

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

    14. Applied associate-*l*6.9

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

    15. Simplified5.8

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.740368709049869308088666682154706200096 \cdot 10^{-97} \lor \neg \left(t \le 2.823511479814487689030883192213886222863 \cdot 10^{-176}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1 z)))) -7.62322630331204244e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.41339449277023022e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))