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

Average Error: 24.5 → 9.1

Time: 16.0s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.153952848057346 \cdot 10^{-251} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\left(x + \frac{z}{a - t} \cdot \left(y - x\right)\right) + \left(y - x\right) \cdot \left(-\frac{\frac{t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r751032 = x;
        double r751033 = y;
        double r751034 = r751033 - r751032;
        double r751035 = z;
        double r751036 = t;
        double r751037 = r751035 - r751036;
        double r751038 = r751034 * r751037;
        double r751039 = a;
        double r751040 = r751039 - r751036;
        double r751041 = r751038 / r751040;
        double r751042 = r751032 + r751041;
        return r751042;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r751043 = x;
        double r751044 = y;
        double r751045 = r751044 - r751043;
        double r751046 = z;
        double r751047 = t;
        double r751048 = r751046 - r751047;
        double r751049 = r751045 * r751048;
        double r751050 = a;
        double r751051 = r751050 - r751047;
        double r751052 = r751049 / r751051;
        double r751053 = r751043 + r751052;
        double r751054 = -2.153952848057346e-251;
        bool r751055 = r751053 <= r751054;
        double r751056 = 0.0;
        bool r751057 = r751053 <= r751056;
        double r751058 = !r751057;
        bool r751059 = r751055 || r751058;
        double r751060 = r751046 / r751051;
        double r751061 = r751060 * r751045;
        double r751062 = r751043 + r751061;
        double r751063 = cbrt(r751051);
        double r751064 = r751063 * r751063;
        double r751065 = r751047 / r751064;
        double r751066 = r751065 / r751063;
        double r751067 = -r751066;
        double r751068 = r751045 * r751067;
        double r751069 = r751062 + r751068;
        double r751070 = r751043 * r751046;
        double r751071 = r751070 / r751047;
        double r751072 = r751044 + r751071;
        double r751073 = r751046 * r751044;
        double r751074 = r751073 / r751047;
        double r751075 = r751072 - r751074;
        double r751076 = r751059 ? r751069 : r751075;
        return r751076;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	9.2
Herbie	9.1

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -2.153952848057346e-251 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.6

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

   3. Applied *-un-lft-identity 21.6

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

   4. Applied times-frac 7.5

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

   5. Simplified 7.5

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

   7. Applied div-sub 7.5

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

   9. Applied sub-neg 7.5

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

   10. Applied distribute-lft-in 7.5

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

   11. Applied associate-+r+ 7.5

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

   12. Simplified 7.5

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

   14. Applied add-cube-cbrt 7.8

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

   15. Applied associate-/r* 7.8

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

   if -2.153952848057346e-251 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 55.8

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

   2. Taylor expanded around inf 22.3

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -2.153952848057346 \cdot 10^{-251} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0\right):\\ \;\;\;\;\left(x + \frac{z}{a - t} \cdot \left(y - x\right)\right) + \left(y - x\right) \cdot \left(-\frac{\frac{t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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