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

Average Error: 23.8 → 9.7

Time: 21.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -3.4167661692054223 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}} \cdot \frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}} + x\\ \mathbf{elif}\;a \le 1.3917837591469568 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}} \cdot \frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r31089257 = x;
        double r31089258 = y;
        double r31089259 = z;
        double r31089260 = r31089258 - r31089259;
        double r31089261 = t;
        double r31089262 = r31089261 - r31089257;
        double r31089263 = r31089260 * r31089262;
        double r31089264 = a;
        double r31089265 = r31089264 - r31089259;
        double r31089266 = r31089263 / r31089265;
        double r31089267 = r31089257 + r31089266;
        return r31089267;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r31089268 = a;
        double r31089269 = -3.4167661692054223e-133;
        bool r31089270 = r31089268 <= r31089269;
        double r31089271 = t;
        double r31089272 = x;
        double r31089273 = r31089271 - r31089272;
        double r31089274 = cbrt(r31089273);
        double r31089275 = z;
        double r31089276 = r31089268 - r31089275;
        double r31089277 = cbrt(r31089276);
        double r31089278 = r31089274 / r31089277;
        double r31089279 = y;
        double r31089280 = r31089279 - r31089275;
        double r31089281 = r31089277 / r31089274;
        double r31089282 = r31089281 * r31089281;
        double r31089283 = r31089280 / r31089282;
        double r31089284 = r31089278 * r31089283;
        double r31089285 = r31089284 + r31089272;
        double r31089286 = 1.3917837591469568e-93;
        bool r31089287 = r31089268 <= r31089286;
        double r31089288 = r31089272 / r31089275;
        double r31089289 = r31089271 / r31089275;
        double r31089290 = r31089288 - r31089289;
        double r31089291 = r31089279 * r31089290;
        double r31089292 = r31089291 + r31089271;
        double r31089293 = r31089287 ? r31089292 : r31089285;
        double r31089294 = r31089270 ? r31089285 : r31089293;
        return r31089294;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	23.8
Target	11.9
Herbie	9.7

\[\begin{array}{l} \mathbf{if}\;z \lt -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -3.4167661692054223e-133 or 1.3917837591469568e-93 < a

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.9

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

   4. Applied times-frac 9.2

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 9.2

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

   7. Applied cbrt-prod 9.2

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

   8. Applied add-cube-cbrt 9.4

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

   9. Applied times-frac 9.4

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

   10. Applied associate-*r* 8.9

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

   11. Simplified 9.1

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

   if -3.4167661692054223e-133 < a < 1.3917837591469568e-93

   1. Initial program 29.7

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

   3. Applied add-cube-cbrt 30.3

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

   4. Applied times-frac 21.6

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 21.6

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

   7. Applied cbrt-prod 21.6

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

   8. Applied add-cube-cbrt 21.8

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

   9. Applied times-frac 21.8

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

   10. Applied associate-*r* 20.8

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

   11. Simplified 20.7

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

   12. Taylor expanded around inf 14.3

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

   13. Simplified 11.2

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -3.4167661692054223 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}} \cdot \frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}} + x\\ \mathbf{elif}\;a \le 1.3917837591469568 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}} \cdot \frac{y - z}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}} \cdot \frac{\sqrt[3]{a - z}}{\sqrt[3]{t - x}}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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