Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 24.9 → 10.5

Time: 21.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.575384619930811018125252150797882880606 \cdot 10^{227}:\\ \;\;\;\;\frac{x}{t} \cdot z - \left(\frac{z}{t} \cdot y - y\right)\\ \mathbf{elif}\;t \le 8.293359549117060522486462892699697431159 \cdot 10^{222}:\\ \;\;\;\;x + \sqrt[3]{\frac{z - t}{a - t}} \cdot \left(\left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \left(y - x\right)\right) \cdot \sqrt[3]{\frac{z - t}{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{\frac{t}{z}}\right) - \frac{z}{\frac{t}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r513954 = x;
        double r513955 = y;
        double r513956 = r513955 - r513954;
        double r513957 = z;
        double r513958 = t;
        double r513959 = r513957 - r513958;
        double r513960 = r513956 * r513959;
        double r513961 = a;
        double r513962 = r513961 - r513958;
        double r513963 = r513960 / r513962;
        double r513964 = r513954 + r513963;
        return r513964;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r513965 = t;
        double r513966 = -2.575384619930811e+227;
        bool r513967 = r513965 <= r513966;
        double r513968 = x;
        double r513969 = r513968 / r513965;
        double r513970 = z;
        double r513971 = r513969 * r513970;
        double r513972 = r513970 / r513965;
        double r513973 = y;
        double r513974 = r513972 * r513973;
        double r513975 = r513974 - r513973;
        double r513976 = r513971 - r513975;
        double r513977 = 8.29335954911706e+222;
        bool r513978 = r513965 <= r513977;
        double r513979 = r513970 - r513965;
        double r513980 = a;
        double r513981 = r513980 - r513965;
        double r513982 = r513979 / r513981;
        double r513983 = cbrt(r513982);
        double r513984 = r513973 - r513968;
        double r513985 = r513983 * r513984;
        double r513986 = r513985 * r513983;
        double r513987 = r513983 * r513986;
        double r513988 = r513968 + r513987;
        double r513989 = r513965 / r513970;
        double r513990 = r513968 / r513989;
        double r513991 = r513973 + r513990;
        double r513992 = r513965 / r513973;
        double r513993 = r513970 / r513992;
        double r513994 = r513991 - r513993;
        double r513995 = r513978 ? r513988 : r513994;
        double r513996 = r513967 ? r513976 : r513995;
        return r513996;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.9
Target	9.4
Herbie	10.5

\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -2.575384619930811e+227

   1. Initial program 51.6

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

   2. Simplified 32.1

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

   4. Applied div-inv 32.2

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

   5. Applied associate-*l* 25.4

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

   6. Simplified 25.3

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

   8. Applied add-cube-cbrt 25.5

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

   9. Applied associate-*r* 25.5

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

   10. Simplified 25.5

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

   11. Taylor expanded around inf 23.2

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

   12. Simplified 13.7

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

   if -2.575384619930811e+227 < t < 8.29335954911706e+222

   1. Initial program 19.7

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

   2. Simplified 11.8

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

   4. Applied div-inv 11.9

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

   5. Applied associate-*l* 9.6

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

   6. Simplified 9.5

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

   8. Applied add-cube-cbrt 10.0

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

   9. Applied associate-*r* 10.0

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

   10. Simplified 10.0

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

   if 8.29335954911706e+222 < t

   1. Initial program 51.5

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

   2. Simplified 32.7

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

   4. Applied div-inv 32.7

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

   5. Applied associate-*l* 26.6

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

   6. Simplified 26.5

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

   8. Applied add-cube-cbrt 26.7

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

   9. Applied associate-*r* 26.7

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

   10. Simplified 26.7

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

   12. Applied add-cube-cbrt 26.8

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

   13. Applied *-un-lft-identity 26.8

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

   14. Applied times-frac 26.8

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

   15. Applied cbrt-prod 27.1

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

   16. Taylor expanded around inf 22.7

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

   17. Simplified 12.8

       \[\leadsto \color{blue}{\left(y + \frac{x}{\frac{t}{z}}\right) - \frac{z}{\frac{t}{y}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.575384619930811018125252150797882880606 \cdot 10^{227}:\\ \;\;\;\;\frac{x}{t} \cdot z - \left(\frac{z}{t} \cdot y - y\right)\\ \mathbf{elif}\;t \le 8.293359549117060522486462892699697431159 \cdot 10^{222}:\\ \;\;\;\;x + \sqrt[3]{\frac{z - t}{a - t}} \cdot \left(\left(\sqrt[3]{\frac{z - t}{a - t}} \cdot \left(y - x\right)\right) \cdot \sqrt[3]{\frac{z - t}{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{\frac{t}{z}}\right) - \frac{z}{\frac{t}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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