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

Average Error: 25.1 → 7.5

Time: 33.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2.9577377713103016 \cdot 10^{-241}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(\frac{a}{\frac{t}{y}} + \left(y + \frac{x}{\frac{t}{z}}\right)\right) - \left(\frac{z}{\frac{t}{y}} + \frac{a}{\frac{t}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\\ \end{array}\]

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 (<= (+ x (/ (* (- y x) (- z t)) (- a t))) -2.9577377713103016e-241)
   (+ x (* (- y x) (/ (- z t) (- a t))))
   (if (<= (+ x (/ (* (- y x) (- z t)) (- a t))) 0.0)
     (- (+ (/ a (/ t y)) (+ y (/ x (/ t z)))) (+ (/ z (/ t y)) (/ a (/ t x))))
     (+
      x
      (*
       (*
        (- y x)
        (/
         (* (cbrt (- z t)) (cbrt (- z t)))
         (* (cbrt (- a t)) (cbrt (- a t)))))
       (/ (cbrt (- z t)) (cbrt (- 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))) <= -2.9577377713103016e-241) {
		tmp = x + ((y - x) * ((z - t) / (a - t)));
	} else if ((x + (((y - x) * (z - t)) / (a - t))) <= 0.0) {
		tmp = ((a / (t / y)) + (y + (x / (t / z)))) - ((z / (t / y)) + (a / (t / x)));
	} else {
		tmp = x + (((y - x) * ((cbrt(z - t) * cbrt(z - t)) / (cbrt(a - t) * cbrt(a - t)))) * (cbrt(z - t) / cbrt(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	25.1
Target	9.2
Herbie	7.5

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2.95773777131030165e-241

   1. Initial program 22.6

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

   2. Simplified 7.3

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

   if -2.95773777131030165e-241 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 54.2

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

   2. Simplified 53.9

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

   3. Taylor expanded around inf 6.9

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

   4. Simplified 10.0

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

   if 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. Simplified 7.1

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

   4. Applied add-cube-cbrt_binary64_11366 7.8

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

   5. Applied add-cube-cbrt_binary64_11366 7.6

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

   6. Applied times-frac_binary64_11337 7.6

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

   7. Applied associate-*r*_binary64_11271 7.1

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

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2.9577377713103016 \cdot 10^{-241}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(\frac{a}{\frac{t}{y}} + \left(y + \frac{x}{\frac{t}{z}}\right)\right) - \left(\frac{z}{\frac{t}{y}} + \frac{a}{\frac{t}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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))))