Average Error: 6.7 → 2.5
Time: 5.8s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.5168010274110233 \cdot 10^{-302} \lor \neg \left(x \le 1.567521217467982 \cdot 10^{132}\right):\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.5168010274110233 \cdot 10^{-302} \lor \neg \left(x \le 1.567521217467982 \cdot 10^{132}\right):\\
\;\;\;\;\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r557666 = x;
        double r557667 = 2.0;
        double r557668 = r557666 * r557667;
        double r557669 = y;
        double r557670 = z;
        double r557671 = r557669 * r557670;
        double r557672 = t;
        double r557673 = r557672 * r557670;
        double r557674 = r557671 - r557673;
        double r557675 = r557668 / r557674;
        return r557675;
}
↓
double f(double x, double y, double z, double t) {
        double r557676 = x;
        double r557677 = -7.516801027411023e-302;
        bool r557678 = r557676 <= r557677;
        double r557679 = 1.567521217467982e+132;
        bool r557680 = r557676 <= r557679;
        double r557681 = !r557680;
        bool r557682 = r557678 || r557681;
        double r557683 = cbrt(r557676);
        double r557684 = cbrt(r557683);
        double r557685 = r557684 * r557684;
        double r557686 = z;
        double r557687 = cbrt(r557686);
        double r557688 = r557687 * r557687;
        double r557689 = r557688 / r557685;
        double r557690 = r557685 / r557689;
        double r557691 = r557687 / r557684;
        double r557692 = r557684 / r557691;
        double r557693 = y;
        double r557694 = t;
        double r557695 = r557693 - r557694;
        double r557696 = 2.0;
        double r557697 = r557695 / r557696;
        double r557698 = r557683 / r557697;
        double r557699 = r557692 * r557698;
        double r557700 = r557690 * r557699;
        double r557701 = r557676 / r557686;
        double r557702 = r557701 / r557697;
        double r557703 = r557682 ? r557700 : r557702;
        return r557703;
}

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

Original6.7
Target2.3
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.516801027411023e-302 or 1.567521217467982e+132 < x

    1. Initial program 8.2

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

    2. Simplified7.2

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

    4. Applied *-un-lft-identity7.2

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

    5. Applied times-frac7.2

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

    6. Applied add-cube-cbrt7.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{z}{1} \cdot \frac{y - t}{2}}\]

    7. Applied times-frac6.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{z}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}}\]

    8. Simplified6.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt6.3

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

    11. Applied add-cube-cbrt6.4

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

    12. Applied times-frac6.4

      \[\leadsto \frac{\sqrt[3]{x}}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\]

    13. Applied add-cube-cbrt6.7

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

    14. Applied times-frac6.7

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

    15. Applied associate-*l*2.5

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

    if -7.516801027411023e-302 < x < 1.567521217467982e+132

    1. Initial program 4.3

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

    2. Simplified3.2

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

    4. Applied *-un-lft-identity3.2

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

    5. Applied times-frac3.2

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

    6. Applied associate-/r*2.6

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

    7. Simplified2.6

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.5168010274110233 \cdot 10^{-302} \lor \neg \left(x \le 1.567521217467982 \cdot 10^{132}\right):\\ \;\;\;\;\frac{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{\sqrt[3]{x}}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))