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

Average Error: 24.8 → 10.5

Time: 8.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.00499911306160384 \cdot 10^{-132}:\\ \;\;\;\;x + \left(\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t - x}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \mathbf{elif}\;a \le 1.3516224103259182 \cdot 10^{-197}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((a <= -5.004999113061604e-132)) {
		VAR = (x + (((((y - z) / (cbrt((a - z)) * cbrt((a - z)))) * ((cbrt((t - x)) * cbrt((t - x))) / (cbrt(cbrt((a - z))) * cbrt(cbrt((a - z)))))) * (cbrt((cbrt((t - x)) * cbrt((t - x)))) / cbrt((cbrt(cbrt((a - z))) * cbrt(cbrt((a - z))))))) * (cbrt(cbrt((t - x))) / cbrt(cbrt(cbrt((a - z)))))));
	} else {
		double VAR_1;
		if ((a <= 1.3516224103259182e-197)) {
			VAR_1 = ((((x * y) / z) + t) - ((t * y) / z));
		} else {
			VAR_1 = (x + ((((y - z) / (cbrt((a - z)) * cbrt((a - z)))) / (cbrt(cbrt((a - z))) * cbrt(cbrt((a - z))))) * ((t - x) / cbrt(cbrt((a - z))))));
		}
		VAR = VAR_1;
	}
	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	24.8
Target	11.8
Herbie	10.5

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

Derivation

1. Split input into 3 regimes

2. if a < -5.004999113061604e-132

   1. Initial program 23.0

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

   3. Applied add-cube-cbrt 23.4

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

   4. Applied times-frac 9.5

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

   6. Applied add-cube-cbrt 9.7

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

   7. Applied add-cube-cbrt 9.8

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

   8. Applied times-frac 9.8

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

   9. Applied associate-*r* 9.4

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

   11. Applied add-cube-cbrt 9.4

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

   12. Applied cbrt-prod 9.4

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

   13. Applied add-cube-cbrt 9.4

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

   14. Applied cbrt-prod 9.4

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

   15. Applied times-frac 9.4

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

   16. Applied associate-*r* 9.4

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

   if -5.004999113061604e-132 < a < 1.3516224103259182e-197

   1. Initial program 29.4

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

   2. Taylor expanded around inf 12.2

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

   if 1.3516224103259182e-197 < a

   1. Initial program 24.0

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

   3. Applied add-cube-cbrt 24.5

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

   4. Applied times-frac 10.4

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

   6. Applied add-cube-cbrt 10.6

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

   7. Applied *-un-lft-identity 10.6

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

   8. Applied times-frac 10.6

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

   9. Applied associate-*r* 10.6

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

   10. Simplified 10.6

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.00499911306160384 \cdot 10^{-132}:\\ \;\;\;\;x + \left(\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t - x}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \mathbf{elif}\;a \le 1.3516224103259182 \cdot 10^{-197}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))