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

Average Error: 7.0 → 3.2

Time: 10.6s

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 r847068 = x;
        double r847069 = y;
        double r847070 = z;
        double r847071 = r847069 * r847070;
        double r847072 = r847071 - r847068;
        double r847073 = t;
        double r847074 = r847073 * r847070;
        double r847075 = r847074 - r847068;
        double r847076 = r847072 / r847075;
        double r847077 = r847068 + r847076;
        double r847078 = 1.0;
        double r847079 = r847068 + r847078;
        double r847080 = r847077 / r847079;
        return r847080;
}

↓

double f(double x, double y, double z, double t) {
        double r847081 = x;
        double r847082 = 1.379014540377949e-298;
        bool r847083 = r847081 <= r847082;
        double r847084 = y;
        double r847085 = z;
        double r847086 = t;
        double r847087 = r847086 * r847085;
        double r847088 = r847087 - r847081;
        double r847089 = r847085 / r847088;
        double r847090 = r847084 * r847089;
        double r847091 = r847081 + r847090;
        double r847092 = 1.0;
        double r847093 = r847081 + r847092;
        double r847094 = r847091 / r847093;
        double r847095 = r847081 / r847088;
        double r847096 = 3.0;
        double r847097 = pow(r847095, r847096);
        double r847098 = cbrt(r847097);
        double r847099 = r847098 / r847093;
        double r847100 = r847094 - r847099;
        double r847101 = 2.740266379468141e-249;
        bool r847102 = r847081 <= r847101;
        double r847103 = r847084 / r847086;
        double r847104 = r847081 + r847103;
        double r847105 = r847104 / r847093;
        double r847106 = cbrt(r847091);
        double r847107 = r847106 * r847106;
        double r847108 = cbrt(r847093);
        double r847109 = r847108 * r847108;
        double r847110 = r847107 / r847109;
        double r847111 = r847106 / r847108;
        double r847112 = r847110 * r847111;
        double r847113 = r847095 / r847093;
        double r847114 = r847112 - r847113;
        double r847115 = r847102 ? r847105 : r847114;
        double r847116 = r847083 ? r847100 : r847115;
        return r847116;
}

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)))