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

Average Error: 23.8 → 10.3

Time: 6.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -4.16782147323413201 \cdot 10^{-141} \lor \neg \left(a \le 2.1530097580240329 \cdot 10^{-165}\right):\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((a <= -4.167821473234132e-141) || !(a <= 2.153009758024033e-165))) {
		VAR = ((double) (x + ((double) (((double) (((double) (((double) (((double) cbrt(((double) (y - z)))) * ((double) cbrt(((double) (y - z)))))) / ((double) cbrt(((double) (a - z)))))) / ((double) sqrt(((double) cbrt(((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))))))) * ((double) (((double) (((double) (((double) cbrt(((double) (y - z)))) / ((double) cbrt(((double) (a - z)))))) / ((double) sqrt(((double) cbrt(((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))))))) * ((double) (((double) (t - x)) / ((double) cbrt(((double) cbrt(((double) (a - z))))))))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (x * y)) / z)) + t)) - ((double) (((double) (t * y)) / z))));
	}
	return VAR;
}

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	12.1
Herbie	10.3

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

Derivation

1. Split input into 2 regimes

2. if a < -4.167821473234132e-141 or 2.153009758024033e-165 < a

   1. Initial program 22.4

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

   3. Applied add-cube-cbrt 22.8

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

   4. Applied times-frac 10.0

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

   6. Applied add-cube-cbrt 10.1

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

   7. Applied cbrt-prod 10.1

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

   8. Applied *-un-lft-identity 10.1

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

   9. Applied times-frac 10.1

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

   10. Applied associate-*r* 9.8

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

   11. Simplified 9.8

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

   13. Applied add-sqr-sqrt 9.8

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

   14. Applied add-cube-cbrt 9.8

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

   15. Applied times-frac 9.8

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

   16. Applied times-frac 9.8

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

   17. Applied associate-*l* 9.6

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

   if -4.167821473234132e-141 < a < 2.153009758024033e-165

   1. Initial program 29.0

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

   2. Taylor expanded around inf 12.9

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.16782147323413201 \cdot 10^{-141} \lor \neg \left(a \le 2.1530097580240329 \cdot 10^{-165}\right):\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))))