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

Average Error: 24.8 → 10.3

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.2785256636554884 \cdot 10^{-145}:\\ \;\;\;\;x + \left(\frac{y}{\frac{a}{z - t} - \frac{t}{z - t}} - \left(\sqrt[3]{\frac{x}{\frac{a}{z - t} - \frac{t}{z - t}}} \cdot \sqrt[3]{\frac{x}{\frac{a}{z - t} - \frac{t}{z - t}}}\right) \cdot \sqrt[3]{\frac{x}{\frac{a}{z - t} - \frac{t}{z - t}}}\right)\\ \mathbf{elif}\;a \le 2.88288138974080523 \cdot 10^{-216}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r643169 = x;
        double r643170 = y;
        double r643171 = r643170 - r643169;
        double r643172 = z;
        double r643173 = t;
        double r643174 = r643172 - r643173;
        double r643175 = r643171 * r643174;
        double r643176 = a;
        double r643177 = r643176 - r643173;
        double r643178 = r643175 / r643177;
        double r643179 = r643169 + r643178;
        return r643179;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r643180 = a;
        double r643181 = -2.2785256636554884e-145;
        bool r643182 = r643180 <= r643181;
        double r643183 = x;
        double r643184 = y;
        double r643185 = z;
        double r643186 = t;
        double r643187 = r643185 - r643186;
        double r643188 = r643180 / r643187;
        double r643189 = r643186 / r643187;
        double r643190 = r643188 - r643189;
        double r643191 = r643184 / r643190;
        double r643192 = r643183 / r643190;
        double r643193 = cbrt(r643192);
        double r643194 = r643193 * r643193;
        double r643195 = r643194 * r643193;
        double r643196 = r643191 - r643195;
        double r643197 = r643183 + r643196;
        double r643198 = 2.8828813897408052e-216;
        bool r643199 = r643180 <= r643198;
        double r643200 = r643183 * r643185;
        double r643201 = r643200 / r643186;
        double r643202 = r643184 + r643201;
        double r643203 = r643185 * r643184;
        double r643204 = r643203 / r643186;
        double r643205 = r643202 - r643204;
        double r643206 = r643184 - r643183;
        double r643207 = r643180 - r643186;
        double r643208 = r643187 / r643207;
        double r643209 = r643206 * r643208;
        double r643210 = r643183 + r643209;
        double r643211 = r643199 ? r643205 : r643210;
        double r643212 = r643182 ? r643197 : r643211;
        return r643212;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	9.2
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if a < -2.2785256636554884e-145

   1. Initial program 24.2

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

   3. Applied associate-/l* 9.7

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

   5. Applied div-sub 9.7

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

   7. Applied div-sub 9.7

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

   9. Applied add-cube-cbrt 9.8

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

   if -2.2785256636554884e-145 < a < 2.8828813897408052e-216

   1. Initial program 30.1

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

   2. Taylor expanded around inf 12.4

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

   if 2.8828813897408052e-216 < a

   1. Initial program 23.1

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

   3. Applied *-un-lft-identity 23.1

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

   4. Applied times-frac 9.8

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

   5. Simplified 9.8

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.2785256636554884 \cdot 10^{-145}:\\ \;\;\;\;x + \left(\frac{y}{\frac{a}{z - t} - \frac{t}{z - t}} - \left(\sqrt[3]{\frac{x}{\frac{a}{z - t} - \frac{t}{z - t}}} \cdot \sqrt[3]{\frac{x}{\frac{a}{z - t} - \frac{t}{z - t}}}\right) \cdot \sqrt[3]{\frac{x}{\frac{a}{z - t} - \frac{t}{z - t}}}\right)\\ \mathbf{elif}\;a \le 2.88288138974080523 \cdot 10^{-216}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))