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

Average Error: 7.0 → 3.2

Time: 10.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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 r832106 = x;
        double r832107 = y;
        double r832108 = z;
        double r832109 = r832107 * r832108;
        double r832110 = r832109 - r832106;
        double r832111 = t;
        double r832112 = r832111 * r832108;
        double r832113 = r832112 - r832106;
        double r832114 = r832110 / r832113;
        double r832115 = r832106 + r832114;
        double r832116 = 1.0;
        double r832117 = r832106 + r832116;
        double r832118 = r832115 / r832117;
        return r832118;
}

↓

double f(double x, double y, double z, double t) {
        double r832119 = x;
        double r832120 = 1.379014540377949e-298;
        bool r832121 = r832119 <= r832120;
        double r832122 = y;
        double r832123 = z;
        double r832124 = t;
        double r832125 = r832124 * r832123;
        double r832126 = r832125 - r832119;
        double r832127 = r832123 / r832126;
        double r832128 = r832122 * r832127;
        double r832129 = r832119 + r832128;
        double r832130 = 1.0;
        double r832131 = r832119 + r832130;
        double r832132 = r832129 / r832131;
        double r832133 = r832119 / r832126;
        double r832134 = 3.0;
        double r832135 = pow(r832133, r832134);
        double r832136 = cbrt(r832135);
        double r832137 = r832136 / r832131;
        double r832138 = r832132 - r832137;
        double r832139 = 2.740266379468141e-249;
        bool r832140 = r832119 <= r832139;
        double r832141 = r832122 / r832124;
        double r832142 = r832119 + r832141;
        double r832143 = r832142 / r832131;
        double r832144 = cbrt(r832129);
        double r832145 = r832144 * r832144;
        double r832146 = cbrt(r832131);
        double r832147 = r832146 * r832146;
        double r832148 = r832145 / r832147;
        double r832149 = r832144 / r832146;
        double r832150 = r832148 * r832149;
        double r832151 = r832133 / r832131;
        double r832152 = r832150 - r832151;
        double r832153 = r832140 ? r832143 : r832152;
        double r832154 = r832121 ? r832138 : r832153;
        return r832154;
}

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)))