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

Average Error: 24.3 → 7.6

Time: 17.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -5.176650421125973 \cdot 10^{-303} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot \left(y \cdot a\right)}{t \cdot t} + \left(\frac{x \cdot z}{t} + \left(\frac{z \cdot \left(x \cdot a\right)}{t \cdot t} + \left(y + \frac{y \cdot a}{t}\right)\right)\right)\right) - \left(\frac{z \cdot \left(y \cdot a\right)}{t \cdot t} + \left(\frac{y \cdot z}{t} + \left(\frac{x \cdot a}{t} + \frac{x \cdot \left(a \cdot a\right)}{t \cdot t}\right)\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (+ x (/ (* (- y x) (- z t)) (- a t))) -5.176650421125973e-303)
         (not (<= (+ x (/ (* (- y x) (- z t)) (- a t))) 0.0)))
   (+ x (/ (- y x) (/ (- a t) (- z t))))
   (-
    (+
     (/ (* a (* y a)) (* t t))
     (+ (/ (* x z) t) (+ (/ (* z (* x a)) (* t t)) (+ y (/ (* y a) t)))))
    (+
     (/ (* z (* y a)) (* t t))
     (+ (/ (* y z) t) (+ (/ (* x a) t) (/ (* x (* a a)) (* t t))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x + (((y - x) * (z - t)) / (a - t))) <= -5.176650421125973e-303) || !((x + (((y - x) * (z - t)) / (a - t))) <= 0.0)) {
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	} else {
		tmp = (((a * (y * a)) / (t * t)) + (((x * z) / t) + (((z * (x * a)) / (t * t)) + (y + ((y * a) / t))))) - (((z * (y * a)) / (t * t)) + (((y * z) / t) + (((x * a) / t) + ((x * (a * a)) / (t * t)))));
	}
	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	24.3
Target	9.6
Herbie	7.6

\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.17665042112597284e-303 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 21.0

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

   3. Applied associate-/l*_binary64_15709 7.7

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

   if -5.17665042112597284e-303 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 60.9

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

   2. Taylor expanded around inf 5.6

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

   3. Simplified 5.6

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

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -5.176650421125973 \cdot 10^{-303} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot \left(y \cdot a\right)}{t \cdot t} + \left(\frac{x \cdot z}{t} + \left(\frac{z \cdot \left(x \cdot a\right)}{t \cdot t} + \left(y + \frac{y \cdot a}{t}\right)\right)\right)\right) - \left(\frac{z \cdot \left(y \cdot a\right)}{t \cdot t} + \left(\frac{y \cdot z}{t} + \left(\frac{x \cdot a}{t} + \frac{x \cdot \left(a \cdot a\right)}{t \cdot t}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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