Average Error: 4.8 → 3.2
Time: 9.0s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3890466172690332:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;z \le 6.52761212898463406 \cdot 10^{48}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{\left(-t\right) \cdot x}{1 - z}\\ \mathbf{elif}\;z \le 4.0178358019584588 \cdot 10^{276}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + t \cdot \frac{-x}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -3890466172690332:\\
\;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\

\mathbf{elif}\;z \le 6.52761212898463406 \cdot 10^{48}:\\
\;\;\;\;\frac{x \cdot y}{z} + \frac{\left(-t\right) \cdot x}{1 - z}\\

\mathbf{elif}\;z \le 4.0178358019584588 \cdot 10^{276}:\\
\;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r491636 = x;
        double r491637 = y;
        double r491638 = z;
        double r491639 = r491637 / r491638;
        double r491640 = t;
        double r491641 = 1.0;
        double r491642 = r491641 - r491638;
        double r491643 = r491640 / r491642;
        double r491644 = r491639 - r491643;
        double r491645 = r491636 * r491644;
        return r491645;
}
↓
double f(double x, double y, double z, double t) {
        double r491646 = z;
        double r491647 = -3890466172690332.0;
        bool r491648 = r491646 <= r491647;
        double r491649 = 1.0;
        double r491650 = x;
        double r491651 = r491649 / r491650;
        double r491652 = y;
        double r491653 = r491646 / r491652;
        double r491654 = r491651 * r491653;
        double r491655 = r491649 / r491654;
        double r491656 = t;
        double r491657 = 1.0;
        double r491658 = r491657 - r491646;
        double r491659 = r491656 / r491658;
        double r491660 = -r491659;
        double r491661 = r491660 * r491650;
        double r491662 = r491655 + r491661;
        double r491663 = 6.527612128984634e+48;
        bool r491664 = r491646 <= r491663;
        double r491665 = r491650 * r491652;
        double r491666 = r491665 / r491646;
        double r491667 = -r491656;
        double r491668 = r491667 * r491650;
        double r491669 = r491668 / r491658;
        double r491670 = r491666 + r491669;
        double r491671 = 4.017835801958459e+276;
        bool r491672 = r491646 <= r491671;
        double r491673 = -r491650;
        double r491674 = r491673 / r491658;
        double r491675 = r491656 * r491674;
        double r491676 = r491666 + r491675;
        double r491677 = r491672 ? r491662 : r491676;
        double r491678 = r491664 ? r491670 : r491677;
        double r491679 = r491648 ? r491662 : r491678;
        return r491679;
}

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

Original4.8
Target4.3
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3890466172690332.0 or 6.527612128984634e+48 < z < 4.017835801958459e+276

    1. Initial program 2.1

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

    3. Applied sub-neg2.1

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

    4. Applied distribute-lft-in2.1

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

    5. Simplified6.4

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

    6. Simplified6.4

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

    8. Applied clear-num6.6

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

    10. Applied *-un-lft-identity6.6

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

    11. Applied times-frac2.3

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

    if -3890466172690332.0 < z < 6.527612128984634e+48

    1. Initial program 7.8

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

    3. Applied sub-neg7.8

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

    4. Applied distribute-lft-in7.8

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

    5. Simplified3.3

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

    6. Simplified3.3

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

    8. Applied distribute-neg-frac3.3

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

    9. Applied associate-*l/3.4

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

    if 4.017835801958459e+276 < z

    1. Initial program 3.0

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

    3. Applied sub-neg3.0

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

    4. Applied distribute-lft-in3.0

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

    5. Simplified10.1

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

    6. Simplified10.1

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

    8. Applied div-inv10.1

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

    9. Applied distribute-rgt-neg-in10.1

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

    10. Applied associate-*l*16.0

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

    11. Simplified16.0

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3890466172690332:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;z \le 6.52761212898463406 \cdot 10^{48}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{\left(-t\right) \cdot x}{1 - z}\\ \mathbf{elif}\;z \le 4.0178358019584588 \cdot 10^{276}:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}} + \left(-\frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + t \cdot \frac{-x}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

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