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

Average Error: 24.6 → 11.6

Time: 16.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.543954312374929 \cdot 10^{-257}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \le 2.94699012045890585 \cdot 10^{-116}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - x}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{z - t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r761044 = x;
        double r761045 = y;
        double r761046 = r761045 - r761044;
        double r761047 = z;
        double r761048 = t;
        double r761049 = r761047 - r761048;
        double r761050 = r761046 * r761049;
        double r761051 = a;
        double r761052 = r761051 - r761048;
        double r761053 = r761050 / r761052;
        double r761054 = r761044 + r761053;
        return r761054;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r761055 = a;
        double r761056 = -3.543954312374929e-257;
        bool r761057 = r761055 <= r761056;
        double r761058 = x;
        double r761059 = y;
        double r761060 = r761059 - r761058;
        double r761061 = z;
        double r761062 = t;
        double r761063 = r761061 - r761062;
        double r761064 = r761055 - r761062;
        double r761065 = r761063 / r761064;
        double r761066 = r761060 * r761065;
        double r761067 = r761058 + r761066;
        double r761068 = 2.946990120458906e-116;
        bool r761069 = r761055 <= r761068;
        double r761070 = r761058 * r761061;
        double r761071 = r761070 / r761062;
        double r761072 = r761059 + r761071;
        double r761073 = r761061 * r761059;
        double r761074 = r761073 / r761062;
        double r761075 = r761072 - r761074;
        double r761076 = cbrt(r761064);
        double r761077 = r761076 * r761076;
        double r761078 = r761060 / r761077;
        double r761079 = r761076 / r761063;
        double r761080 = r761078 / r761079;
        double r761081 = r761058 + r761080;
        double r761082 = r761069 ? r761075 : r761081;
        double r761083 = r761057 ? r761067 : r761082;
        return r761083;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.6
Target	9.3
Herbie	11.6

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

Derivation

1. Split input into 3 regimes

2. if a < -3.543954312374929e-257

   1. Initial program 23.5

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

   3. Applied *-un-lft-identity 23.5

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

   4. Applied times-frac 11.5

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

   5. Simplified 11.5

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

   if -3.543954312374929e-257 < a < 2.946990120458906e-116

   1. Initial program 29.9

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

   2. Taylor expanded around inf 14.8

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

   if 2.946990120458906e-116 < a

   1. Initial program 23.7

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

   3. Applied associate-/l* 8.8

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

   5. Applied *-un-lft-identity 8.8

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

   6. Applied add-cube-cbrt 9.4

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

   7. Applied times-frac 9.4

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

   8. Applied associate-/r* 10.3

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

   9. Simplified 10.3

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

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.543954312374929 \cdot 10^{-257}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \le 2.94699012045890585 \cdot 10^{-116}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - x}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))