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

Average Error: 24.9 → 6.8

Time: 17.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2.9892933670624995 \cdot 10^{-304} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{t} + \left(y + \frac{x \cdot z}{t}\right)\right) - \left(\frac{y \cdot z}{t} + \frac{x \cdot a}{t}\right)\\ \end{array}\]

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))) -2.9892933670624995e-304)
         (not (<= (+ x (/ (* (- y x) (- z t)) (- a t))) 0.0)))
   (+
    x
    (*
     (*
      (- y x)
      (/ (* (cbrt (- z t)) (cbrt (- z t))) (* (cbrt (- a t)) (cbrt (- a t)))))
     (/ (cbrt (- z t)) (cbrt (- a t)))))
   (- (+ (/ (* y a) t) (+ y (/ (* x z) t))) (+ (/ (* y z) t) (/ (* x a) 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))) <= -2.9892933670624995e-304) || !((x + (((y - x) * (z - t)) / (a - t))) <= 0.0)) {
		tmp = x + (((y - x) * ((cbrt(z - t) * cbrt(z - t)) / (cbrt(a - t) * cbrt(a - t)))) * (cbrt(z - t) / cbrt(a - t)));
	} else {
		tmp = (((y * a) / t) + (y + ((x * z) / t))) - (((y * z) / t) + ((x * a) / 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	24.9
Target	9.5
Herbie	6.8

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2.9892933670624995e-304 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 21.7

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

   3. Applied *-un-lft-identity_binary64_15082 21.7

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

   4. Applied times-frac_binary64_15088 7.6

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

   5. Simplified 7.6

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

   7. Applied add-cube-cbrt_binary64_15117 8.3

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

   8. Applied add-cube-cbrt_binary64_15117 8.1

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

   9. Applied times-frac_binary64_15088 8.1

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

   10. Applied associate-*r*_binary64_15022 7.4

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

   if -2.9892933670624995e-304 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 61.2

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2.9892933670624995 \cdot 10^{-304} \lor \neg \left(x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0\right):\\ \;\;\;\;x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y \cdot a}{t} + \left(y + \frac{x \cdot z}{t}\right)\right) - \left(\frac{y \cdot z}{t} + \frac{x \cdot a}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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))))