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

Average Error: 25.0 → 6.7

Time: 15.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.2104065643874172 \cdot 10^{-306} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot \left(x \cdot y\right)}{z \cdot z} + \left(\frac{t \cdot a}{z} + \left(t + \frac{x \cdot y}{z}\right)\right)\right) - \left(\frac{x \cdot a}{z} + \left(\frac{y \cdot \left(t \cdot a\right)}{z \cdot z} + \frac{y \cdot t}{z}\right)\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) (- t x)) (- a z))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (+ x (/ (* (- y z) (- t x)) (- a z))) -1.2104065643874172e-306)
         (not (<= (+ x (/ (* (- y z) (- t x)) (- a z))) 0.0)))
   (+ x (* (- t x) (/ (- y z) (- a z))))
   (-
    (+ (/ (* a (* x y)) (* z z)) (+ (/ (* t a) z) (+ t (/ (* x y) z))))
    (+ (/ (* x a) z) (+ (/ (* y (* t a)) (* z z)) (/ (* y t) z))))))

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 tmp;
	if (((x + (((y - z) * (t - x)) / (a - z))) <= -1.2104065643874172e-306) || !((x + (((y - z) * (t - x)) / (a - z))) <= 0.0)) {
		tmp = x + ((t - x) * ((y - z) / (a - z)));
	} else {
		tmp = (((a * (x * y)) / (z * z)) + (((t * a) / z) + (t + ((x * y) / z)))) - (((x * a) / z) + (((y * (t * a)) / (z * z)) + ((y * t) / z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.0
Target	12.3
Herbie	6.7

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.21040656438741722e-306 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 21.6

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

   3. Applied add-cube-cbrt_binary64_14094 22.1

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

   4. Applied times-frac_binary64_14065 8.4

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

   6. Applied add-cube-cbrt_binary64_14094 8.6

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

   7. Applied *-un-lft-identity_binary64_14059 8.6

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

   8. Applied times-frac_binary64_14065 8.6

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

   9. Applied associate-*r*_binary64_13999 8.1

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

   10. Simplified 8.1

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

   12. Applied pow1_binary64_14120 8.1

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

   13. Applied pow1_binary64_14120 8.1

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

   14. Applied pow-prod-down_binary64_14130 8.1

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

   15. Simplified 8.2

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

   17. Applied *-un-lft-identity_binary64_14059 8.2

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

   18. Applied *-un-lft-identity_binary64_14059 8.2

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

   19. Applied distribute-lft-out_binary64_14010 8.2

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

   20. Simplified 7.2

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

   if -1.21040656438741722e-306 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 61.1

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

   2. Taylor expanded around 0 1.8

      \[\leadsto \color{blue}{\left(\frac{a \cdot \left(x \cdot y\right)}{{z}^{2}} + \left(\frac{a \cdot t}{z} + \left(t + \frac{x \cdot y}{z}\right)\right)\right) - \left(\frac{a \cdot x}{z} + \left(\frac{a \cdot \left(y \cdot t\right)}{{z}^{2}} + \frac{t \cdot y}{z}\right)\right)}\]

   3. Simplified 1.8

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.2104065643874172 \cdot 10^{-306} \lor \neg \left(x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0\right):\\ \;\;\;\;x + \left(t - x\right) \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot \left(x \cdot y\right)}{z \cdot z} + \left(\frac{t \cdot a}{z} + \left(t + \frac{x \cdot y}{z}\right)\right)\right) - \left(\frac{x \cdot a}{z} + \left(\frac{y \cdot \left(t \cdot a\right)}{z \cdot z} + \frac{y \cdot t}{z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(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))))