Average Error: 7.0 → 3.1
Time: 6.8s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.7183563329919357 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{y - t}{2}} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \le 8.1507283132915605 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{\frac{1 \cdot \left(z \cdot \left(y - t\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y - t}{2}}{x}}}{z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.7183563329919357 \cdot 10^{77}:\\
\;\;\;\;\frac{1}{\frac{y - t}{2}} \cdot \frac{x}{z}\\

\mathbf{elif}\;z \le 8.1507283132915605 \cdot 10^{-182}:\\
\;\;\;\;\frac{x}{\frac{1 \cdot \left(z \cdot \left(y - t\right)\right)}{2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{y - t}{2}}{x}}}{z}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r504293 = x;
        double r504294 = 2.0;
        double r504295 = r504293 * r504294;
        double r504296 = y;
        double r504297 = z;
        double r504298 = r504296 * r504297;
        double r504299 = t;
        double r504300 = r504299 * r504297;
        double r504301 = r504298 - r504300;
        double r504302 = r504295 / r504301;
        return r504302;
}
↓
double f(double x, double y, double z, double t) {
        double r504303 = z;
        double r504304 = -2.7183563329919357e+77;
        bool r504305 = r504303 <= r504304;
        double r504306 = 1.0;
        double r504307 = y;
        double r504308 = t;
        double r504309 = r504307 - r504308;
        double r504310 = 2.0;
        double r504311 = r504309 / r504310;
        double r504312 = r504306 / r504311;
        double r504313 = x;
        double r504314 = r504313 / r504303;
        double r504315 = r504312 * r504314;
        double r504316 = 8.15072831329156e-182;
        bool r504317 = r504303 <= r504316;
        double r504318 = r504303 * r504309;
        double r504319 = r504306 * r504318;
        double r504320 = r504319 / r504310;
        double r504321 = r504313 / r504320;
        double r504322 = r504311 / r504313;
        double r504323 = r504306 / r504322;
        double r504324 = r504323 / r504303;
        double r504325 = r504317 ? r504321 : r504324;
        double r504326 = r504305 ? r504315 : r504325;
        return r504326;
}

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

Original7.0
Target2.1
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.7183563329919357e+77

    1. Initial program 12.9

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified10.5

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

    4. Applied *-un-lft-identity10.5

      \[\leadsto \frac{x}{\frac{z \cdot \left(y - t\right)}{\color{blue}{1 \cdot 2}}}\]

    5. Applied times-frac10.5

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{y - t}{2}}}\]

    6. Applied *-un-lft-identity10.5

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{z}{1} \cdot \frac{y - t}{2}}\]

    7. Applied times-frac2.1

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{1}} \cdot \frac{x}{\frac{y - t}{2}}}\]

    8. Simplified2.1

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
    9. Using strategy rm

    10. Applied associate-*l/2.0

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{\frac{y - t}{2}}}{z}}\]

    11. Simplified2.0

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y - t}{2}}}}{z}\]
    12. Using strategy rm

    13. Applied clear-num2.1

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{y - t}{2}}{x}}}}{z}\]
    14. Using strategy rm

    15. Applied *-un-lft-identity2.1

      \[\leadsto \frac{\frac{1}{\frac{\frac{y - t}{2}}{x}}}{\color{blue}{1 \cdot z}}\]

    16. Applied div-inv2.1

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{y - t}{2} \cdot \frac{1}{x}}}}{1 \cdot z}\]

    17. Applied add-sqr-sqrt2.1

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{y - t}{2} \cdot \frac{1}{x}}}{1 \cdot z}\]

    18. Applied times-frac2.1

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\frac{y - t}{2}} \cdot \frac{\sqrt{1}}{\frac{1}{x}}}}{1 \cdot z}\]

    19. Applied times-frac2.1

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\frac{y - t}{2}}}{1} \cdot \frac{\frac{\sqrt{1}}{\frac{1}{x}}}{z}}\]

    20. Simplified2.1

      \[\leadsto \color{blue}{\frac{1}{\frac{y - t}{2}}} \cdot \frac{\frac{\sqrt{1}}{\frac{1}{x}}}{z}\]

    21. Simplified2.1

      \[\leadsto \frac{1}{\frac{y - t}{2}} \cdot \color{blue}{\frac{x}{z}}\]

    if -2.7183563329919357e+77 < z < 8.15072831329156e-182

    1. Initial program 3.1

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified3.0

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

    4. Applied *-un-lft-identity3.0

      \[\leadsto \frac{x}{\frac{\color{blue}{1 \cdot \left(z \cdot \left(y - t\right)\right)}}{2}}\]

    if 8.15072831329156e-182 < z

    1. Initial program 7.6

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified5.9

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

    4. Applied *-un-lft-identity5.9

      \[\leadsto \frac{x}{\frac{z \cdot \left(y - t\right)}{\color{blue}{1 \cdot 2}}}\]

    5. Applied times-frac5.9

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{y - t}{2}}}\]

    6. Applied *-un-lft-identity5.9

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{z}{1} \cdot \frac{y - t}{2}}\]

    7. Applied times-frac3.4

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{1}} \cdot \frac{x}{\frac{y - t}{2}}}\]

    8. Simplified3.4

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
    9. Using strategy rm

    10. Applied associate-*l/3.4

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{\frac{y - t}{2}}}{z}}\]

    11. Simplified3.4

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y - t}{2}}}}{z}\]
    12. Using strategy rm

    13. Applied clear-num3.6

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{y - t}{2}}{x}}}}{z}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.7183563329919357 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{y - t}{2}} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \le 8.1507283132915605 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{\frac{1 \cdot \left(z \cdot \left(y - t\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y - t}{2}}{x}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))