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

Average Error: 24.3 → 9.6

Time: 18.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -8.572794558051977100123683312677994148346 \cdot 10^{-160} \lor \neg \left(a \le 9.308003628992163531527319396241615875898 \cdot 10^{-174}\right):\\ \;\;\;\;\frac{y - x}{\frac{1}{\frac{z - t}{a - t}}} + x\\ \mathbf{else}:\\ \;\;\;\;y + \left(\frac{x}{\frac{t}{z}} - \frac{z}{\frac{t}{y}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r553038 = x;
        double r553039 = y;
        double r553040 = r553039 - r553038;
        double r553041 = z;
        double r553042 = t;
        double r553043 = r553041 - r553042;
        double r553044 = r553040 * r553043;
        double r553045 = a;
        double r553046 = r553045 - r553042;
        double r553047 = r553044 / r553046;
        double r553048 = r553038 + r553047;
        return r553048;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r553049 = a;
        double r553050 = -8.572794558051977e-160;
        bool r553051 = r553049 <= r553050;
        double r553052 = 9.308003628992164e-174;
        bool r553053 = r553049 <= r553052;
        double r553054 = !r553053;
        bool r553055 = r553051 || r553054;
        double r553056 = y;
        double r553057 = x;
        double r553058 = r553056 - r553057;
        double r553059 = 1.0;
        double r553060 = z;
        double r553061 = t;
        double r553062 = r553060 - r553061;
        double r553063 = r553049 - r553061;
        double r553064 = r553062 / r553063;
        double r553065 = r553059 / r553064;
        double r553066 = r553058 / r553065;
        double r553067 = r553066 + r553057;
        double r553068 = r553061 / r553060;
        double r553069 = r553057 / r553068;
        double r553070 = r553061 / r553056;
        double r553071 = r553060 / r553070;
        double r553072 = r553069 - r553071;
        double r553073 = r553056 + r553072;
        double r553074 = r553055 ? r553067 : r553073;
        return r553074;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.3
Target	9.2
Herbie	9.6

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

Derivation

1. Split input into 2 regimes

2. if a < -8.572794558051977e-160 or 9.308003628992164e-174 < a

   1. Initial program 23.0

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

   2. Simplified 12.1

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

   4. Applied div-inv 12.2

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

   5. Applied associate-*l* 9.4

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

   6. Simplified 9.3

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

   8. Applied *-un-lft-identity 9.3

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

   9. Applied associate-*l* 9.3

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

   10. Simplified 9.3

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

   12. Applied clear-num 9.4

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

   if -8.572794558051977e-160 < a < 9.308003628992164e-174

   1. Initial program 29.5

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

   2. Simplified 24.5

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

   4. Applied div-inv 24.6

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

   5. Applied associate-*l* 19.9

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

   6. Simplified 19.8

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

   8. Applied *-un-lft-identity 19.8

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

   9. Applied associate-*l* 19.8

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

   10. Simplified 19.8

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

   12. Applied clear-num 19.8

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

   13. Taylor expanded around inf 12.5

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

   14. Simplified 10.4

       \[\leadsto \color{blue}{y + \left(\frac{x}{\frac{t}{z}} - \frac{z}{\frac{t}{y}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.572794558051977100123683312677994148346 \cdot 10^{-160} \lor \neg \left(a \le 9.308003628992163531527319396241615875898 \cdot 10^{-174}\right):\\ \;\;\;\;\frac{y - x}{\frac{1}{\frac{z - t}{a - t}}} + x\\ \mathbf{else}:\\ \;\;\;\;y + \left(\frac{x}{\frac{t}{z}} - \frac{z}{\frac{t}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))