Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.0 → 3.2

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.3790145403779491 \cdot 10^{-298}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x + 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\ \mathbf{elif}\;x \le 2.74026637946814113 \cdot 10^{-249}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y \cdot \frac{z}{t \cdot z - x}} \cdot \sqrt[3]{x + y \cdot \frac{z}{t \cdot z - x}}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{x + y \cdot \frac{z}{t \cdot z - x}}}{\sqrt[3]{x + 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r703786 = x;
        double r703787 = y;
        double r703788 = z;
        double r703789 = r703787 * r703788;
        double r703790 = r703789 - r703786;
        double r703791 = t;
        double r703792 = r703791 * r703788;
        double r703793 = r703792 - r703786;
        double r703794 = r703790 / r703793;
        double r703795 = r703786 + r703794;
        double r703796 = 1.0;
        double r703797 = r703786 + r703796;
        double r703798 = r703795 / r703797;
        return r703798;
}

↓

double f(double x, double y, double z, double t) {
        double r703799 = x;
        double r703800 = 1.379014540377949e-298;
        bool r703801 = r703799 <= r703800;
        double r703802 = y;
        double r703803 = z;
        double r703804 = t;
        double r703805 = r703804 * r703803;
        double r703806 = r703805 - r703799;
        double r703807 = r703803 / r703806;
        double r703808 = r703802 * r703807;
        double r703809 = r703799 + r703808;
        double r703810 = 1.0;
        double r703811 = r703799 + r703810;
        double r703812 = r703809 / r703811;
        double r703813 = r703799 / r703806;
        double r703814 = 3.0;
        double r703815 = pow(r703813, r703814);
        double r703816 = cbrt(r703815);
        double r703817 = r703816 / r703811;
        double r703818 = r703812 - r703817;
        double r703819 = 2.740266379468141e-249;
        bool r703820 = r703799 <= r703819;
        double r703821 = r703802 / r703804;
        double r703822 = r703799 + r703821;
        double r703823 = r703822 / r703811;
        double r703824 = cbrt(r703809);
        double r703825 = r703824 * r703824;
        double r703826 = cbrt(r703811);
        double r703827 = r703826 * r703826;
        double r703828 = r703825 / r703827;
        double r703829 = r703824 / r703826;
        double r703830 = r703828 * r703829;
        double r703831 = r703813 / r703811;
        double r703832 = r703830 - r703831;
        double r703833 = r703820 ? r703823 : r703832;
        double r703834 = r703801 ? r703818 : r703833;
        return r703834;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	0.4
Herbie	3.2

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

Derivation

1. Split input into 3 regimes

2. if x < 1.379014540377949e-298

   1. Initial program 6.7

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 6.7

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

   4. Applied associate-+r- 6.7

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

   5. Applied div-sub 6.7

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

   7. Applied *-un-lft-identity 6.7

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

   8. Applied times-frac 2.1

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

   9. Simplified 2.1

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

   11. Applied add-cbrt-cube 9.0

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

   12. Applied add-cbrt-cube 30.7

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

   13. Applied cbrt-undiv 30.7

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

   14. Simplified 2.9

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

   if 1.379014540377949e-298 < x < 2.740266379468141e-249

   1. Initial program 10.4

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

   2. Taylor expanded around inf 15.3

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

   if 2.740266379468141e-249 < x

   1. Initial program 7.1

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 7.1

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

   4. Applied associate-+r- 7.1

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

   5. Applied div-sub 7.1

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

   7. Applied *-un-lft-identity 7.1

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

   8. Applied times-frac 2.0

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

   9. Simplified 2.0

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

   11. Applied add-cube-cbrt 2.8

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

   12. Applied add-cube-cbrt 2.5

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

   13. Applied times-frac 2.5

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.3790145403779491 \cdot 10^{-298}:\\ \;\;\;\;\frac{x + y \cdot \frac{z}{t \cdot z - x}}{x + 1} - \frac{\sqrt[3]{{\left(\frac{x}{t \cdot z - x}\right)}^{3}}}{x + 1}\\ \mathbf{elif}\;x \le 2.74026637946814113 \cdot 10^{-249}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y \cdot \frac{z}{t \cdot z - x}} \cdot \sqrt[3]{x + y \cdot \frac{z}{t \cdot z - x}}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{\sqrt[3]{x + y \cdot \frac{z}{t \cdot z - x}}}{\sqrt[3]{x + 1}} - \frac{\frac{x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

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