Average Error: 7.0 → 2.0
Time: 7.8s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\frac{\frac{x}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}}}}{\frac{\sqrt[3]{z}}{\frac{\sqrt{2}}{\sqrt[3]{y - t}}}}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\frac{\frac{x}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}}}}{\frac{\sqrt[3]{z}}{\frac{\sqrt{2}}{\sqrt[3]{y - t}}}}
double f(double x, double y, double z, double t) {
        double r743762 = x;
        double r743763 = 2.0;
        double r743764 = r743762 * r743763;
        double r743765 = y;
        double r743766 = z;
        double r743767 = r743765 * r743766;
        double r743768 = t;
        double r743769 = r743768 * r743766;
        double r743770 = r743767 - r743769;
        double r743771 = r743764 / r743770;
        return r743771;
}
↓
double f(double x, double y, double z, double t) {
        double r743772 = x;
        double r743773 = z;
        double r743774 = cbrt(r743773);
        double r743775 = r743774 * r743774;
        double r743776 = 2.0;
        double r743777 = sqrt(r743776);
        double r743778 = y;
        double r743779 = t;
        double r743780 = r743778 - r743779;
        double r743781 = cbrt(r743780);
        double r743782 = r743781 * r743781;
        double r743783 = r743777 / r743782;
        double r743784 = r743775 / r743783;
        double r743785 = r743772 / r743784;
        double r743786 = r743777 / r743781;
        double r743787 = r743774 / r743786;
        double r743788 = r743785 / r743787;
        return r743788;
}

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
Target2.2
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 7.0

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

  2. Simplified5.8

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

  4. Applied associate-/l*5.8

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

  6. Applied add-cube-cbrt6.4

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

  7. Applied add-sqr-sqrt6.5

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

  8. Applied times-frac6.5

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

  9. Applied add-cube-cbrt6.7

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

  10. Applied times-frac6.7

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

  11. Applied associate-/r*2.0

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

  12. Final simplification2.0

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

Reproduce

herbie shell --seed 2020025 +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))))