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

Average Error: 25.2 → 8.9

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -6.35262688856634336 \cdot 10^{-298} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 9.92755 \cdot 10^{-256}\right):\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r741985 = x;
        double r741986 = y;
        double r741987 = z;
        double r741988 = r741986 - r741987;
        double r741989 = t;
        double r741990 = r741989 - r741985;
        double r741991 = r741988 * r741990;
        double r741992 = a;
        double r741993 = r741992 - r741987;
        double r741994 = r741991 / r741993;
        double r741995 = r741985 + r741994;
        return r741995;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r741996 = x;
        double r741997 = y;
        double r741998 = z;
        double r741999 = r741997 - r741998;
        double r742000 = t;
        double r742001 = r742000 - r741996;
        double r742002 = r741999 * r742001;
        double r742003 = a;
        double r742004 = r742003 - r741998;
        double r742005 = r742002 / r742004;
        double r742006 = r741996 + r742005;
        double r742007 = -6.352626888566343e-298;
        bool r742008 = r742006 <= r742007;
        double r742009 = 9.92755017398561e-256;
        bool r742010 = r742006 <= r742009;
        double r742011 = !r742010;
        bool r742012 = r742008 || r742011;
        double r742013 = cbrt(r742001);
        double r742014 = r742013 * r742013;
        double r742015 = cbrt(r742004);
        double r742016 = r742015 * r742015;
        double r742017 = r742014 / r742016;
        double r742018 = r741999 * r742017;
        double r742019 = r742013 / r742015;
        double r742020 = r742018 * r742019;
        double r742021 = r741996 + r742020;
        double r742022 = r741996 * r741997;
        double r742023 = r742022 / r741998;
        double r742024 = r742023 + r742000;
        double r742025 = r742000 * r741997;
        double r742026 = r742025 / r741998;
        double r742027 = r742024 - r742026;
        double r742028 = r742012 ? r742021 : r742027;
        return r742028;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.2
Target	12.0
Herbie	8.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -6.352626888566343e-298 or 9.92755017398561e-256 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program 22.0

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

   3. Applied *-un-lft-identity 22.0

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

   4. Applied times-frac 10.5

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

   5. Simplified 10.5

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

   7. Applied add-cube-cbrt 11.1

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

   8. Applied add-cube-cbrt 11.2

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

   9. Applied times-frac 11.2

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

   10. Applied associate-*r* 7.7

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

   if -6.352626888566343e-298 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 9.92755017398561e-256

   1. Initial program 56.7

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

   2. Taylor expanded around inf 20.3

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le -6.35262688856634336 \cdot 10^{-298} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \le 9.92755 \cdot 10^{-256}\right):\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :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))))