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

Average Error: 24.6 → 8.6

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -\infty:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -7.156896122525125 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 6.765930309308169 \cdot 10^{+281}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))) <= ((double) -(((double) INFINITY))))) {
		VAR = ((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) * (1.0 / ((double) (a - z)))))))));
	} else {
		double VAR_1;
		if ((((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))) <= -7.156896122525125e-270)) {
			VAR_1 = ((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))));
		} else {
			double VAR_2;
			if ((((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))) <= 0.0)) {
				VAR_2 = ((double) (t + ((double) (y * ((double) ((x / z) - (t / z)))))));
			} else {
				double VAR_3;
				if ((((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z))))) <= 6.765930309308169e+281)) {
					VAR_3 = ((double) (x + (((double) (((double) (y - z)) * ((double) (t - x)))) / ((double) (a - z)))));
				} else {
					VAR_3 = ((double) (x + ((double) (((double) (y - z)) * ((double) (((double) (t - x)) * (1.0 / ((double) (a - z)))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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	24.6
Target	12.1
Herbie	8.6

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y z) (- t x)) (- a z))) < -inf.0 or 6.7659303093081686e281 < (+ x (/ (* (- y z) (- t x)) (- a z)))

   1. Initial program Error: 61.2 bits

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

   2. Simplified Error: 18.0 bits

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

   4. Applied div-inv Error: 18.1 bits

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

   if -inf.0 < (+ x (/ (* (- y z) (- t x)) (- a z))) < -7.15689612252512464e-270 or 0.0 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 6.7659303093081686e281

   1. Initial program Error: 2.0 bits

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

   if -7.15689612252512464e-270 < (+ x (/ (* (- y z) (- t x)) (- a z))) < 0.0

   1. Initial program Error: 58.2 bits

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

   2. Simplified Error: 58.8 bits

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

   3. Taylor expanded around inf Error: 20.1 bits

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

   4. Simplified Error: 22.7 bits

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

4. Final simplification Error: 8.6 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -\infty:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -7.156896122525125 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 6.765930309308169 \cdot 10^{+281}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))