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

Average Error: 24.6 → 6.5

Time: 22.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.1801785172771272 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \frac{t}{a - z} + \left(\left(x + x \cdot \frac{z}{a - z}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{x \cdot a}{z} + \frac{y \cdot t}{z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 1.1485762430674559 \cdot 10^{+306}:\\ \;\;\;\;\left(\frac{y \cdot t}{a - z} + \left(x + \frac{x \cdot z}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z} - \left(\left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right) + x \cdot \left(\frac{a}{z} + \frac{a}{z} \cdot \frac{a}{z}\right)\right)\\ \end{array}\]

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 (<= (+ x (/ (* (- y z) (- t x)) (- a z))) -1.1801785172771272e-262)
   (+
    (* y (/ t (- a z)))
    (- (+ x (* x (/ z (- a z)))) (+ (* z (/ t (- a z))) (* x (/ y (- a z))))))
   (if (<= (+ x (/ (* (- y z) (- t x)) (- a z))) 0.0)
     (- (+ t (+ (/ (* x y) z) (/ (* t a) z))) (+ (/ (* x a) z) (/ (* y t) z)))
     (if (<= (+ x (/ (* (- y z) (- t x)) (- a z))) 1.1485762430674559e+306)
       (-
        (+ (/ (* y t) (- a z)) (+ x (/ (* x z) (- a z))))
        (+ (/ (* x y) (- a z)) (/ (* z t) (- a z))))
       (-
        (* y (/ t (- a z)))
        (+
         (+ (* z (/ t (- a z))) (* x (/ y (- a z))))
         (* x (+ (/ a z) (* (/ a z) (/ a 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))) <= -1.1801785172771272e-262) {
		tmp = (y * (t / (a - z))) + ((x + (x * (z / (a - z)))) - ((z * (t / (a - z))) + (x * (y / (a - z)))));
	} else if ((x + (((y - z) * (t - x)) / (a - z))) <= 0.0) {
		tmp = (t + (((x * y) / z) + ((t * a) / z))) - (((x * a) / z) + ((y * t) / z));
	} else if ((x + (((y - z) * (t - x)) / (a - z))) <= 1.1485762430674559e+306) {
		tmp = (((y * t) / (a - z)) + (x + ((x * z) / (a - z)))) - (((x * y) / (a - z)) + ((z * t) / (a - z)));
	} else {
		tmp = (y * (t / (a - z))) - (((z * (t / (a - z))) + (x * (y / (a - z)))) + (x * ((a / z) + ((a / z) * (a / 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	24.6
Target	12.1
Herbie	6.5

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

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.1801785172771272e-262

   1. Initial program 21.1

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

   2. Simplified 8.0

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

   3. Taylor expanded around 0 20.9

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

   4. Simplified 6.6

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

   if -1.1801785172771272e-262 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 58.2

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

   2. Simplified 57.7

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

   3. Taylor expanded around inf 3.8

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.14857624306745587e306

   1. Initial program 1.9

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

   2. Simplified 2.8

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

   3. Taylor expanded around 0 1.4

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

   if 1.14857624306745587e306 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 63.7

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

   2. Simplified 17.1

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

   3. Taylor expanded around 0 63.7

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

   4. Simplified 7.3

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

   5. Taylor expanded around inf 30.6

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

   6. Simplified 18.8

      \[\leadsto \frac{t}{a - z} \cdot y + \left(\color{blue}{\left(-x \cdot \left(\frac{a}{z} + \frac{a}{z} \cdot \frac{a}{z}\right)\right)} - \left(\frac{t}{a - z} \cdot z + \frac{y}{a - z} \cdot x\right)\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq -1.1801785172771272 \cdot 10^{-262}:\\ \;\;\;\;y \cdot \frac{t}{a - z} + \left(\left(x + x \cdot \frac{z}{a - z}\right) - \left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right)\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 0:\\ \;\;\;\;\left(t + \left(\frac{x \cdot y}{z} + \frac{t \cdot a}{z}\right)\right) - \left(\frac{x \cdot a}{z} + \frac{y \cdot t}{z}\right)\\ \mathbf{elif}\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \leq 1.1485762430674559 \cdot 10^{+306}:\\ \;\;\;\;\left(\frac{y \cdot t}{a - z} + \left(x + \frac{x \cdot z}{a - z}\right)\right) - \left(\frac{x \cdot y}{a - z} + \frac{z \cdot t}{a - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z} - \left(\left(z \cdot \frac{t}{a - z} + x \cdot \frac{y}{a - z}\right) + x \cdot \left(\frac{a}{z} + \frac{a}{z} \cdot \frac{a}{z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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))))