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

Average Error: 24.3 → 7.2

Time: 18.3s

Precision: binary64

Cost: 3464

Math

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

↓

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

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))) -5.176650421125973e-303)
         (not (<= (+ x (/ (* (- y x) (- z t)) (- a t))) 0.0)))
   (+ x (/ (- y x) (/ (- a t) (- z t))))
   (- (+ (+ y (/ (* x z) t)) (/ (* y a) 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))) <= -5.176650421125973e-303) || !((x + (((y - x) * (z - t)) / (a - t))) <= 0.0)) {
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	} else {
		tmp = ((y + ((x * z) / t)) + ((y * a) / 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	24.3
Target	9.6
Herbie	7.2

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

Alternatives

Alternative 1
Error	8.7
Cost	2696

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

Alternative 2
Error	8.7
Cost	2696

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

Alternative 3
Error	10.9
Cost	2696

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

Alternative 4
Error	18.5
Cost	2181

\[\begin{array}{l} \mathbf{if}\;t \leq -2.243832669059716 \cdot 10^{+216}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;t \leq -5.4161300115267595 \cdot 10^{-177}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 3.3990608062305716 \cdot 10^{-41}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z}}\\ \mathbf{elif}\;t \leq 1.355231778736394 \cdot 10^{+106}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 7.993925956436287 \cdot 10^{+209}:\\ \;\;\;\;z \cdot \frac{y - x}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Alternative 5
Error	18.7
Cost	1988

\[\begin{array}{l} \mathbf{if}\;t \leq -1.862483285948315 \cdot 10^{+66}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq -3.3822985341725177 \cdot 10^{-167}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;t \leq 6.999976732147868 \cdot 10^{-293}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z - t}}\\ \mathbf{elif}\;t \leq 3.15615873131367 \cdot 10^{+87}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \end{array}\]

Alternative 6
Error	22.3
Cost	1674

\[\begin{array}{l} \mathbf{if}\;y \leq -2.436483155066167 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;y \leq 2.1928223815558746 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z}}\\ \mathbf{elif}\;y \leq 4.286444712015001 \cdot 10^{+62} \lor \neg \left(y \leq 1.4457588227513382 \cdot 10^{+128}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a - t}\\ \end{array}\]

Alternative 7
Error	22.5
Cost	1988

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3300816867749887 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;y \leq 1.4074725947547314 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;y \leq 1.2430453641021008 \cdot 10^{+61}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;y \leq 1.2305047984190505 \cdot 10^{+128}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Alternative 8
Error	21.6
Cost	1667

\[\begin{array}{l} \mathbf{if}\;a \leq -5.8095817005789545 \cdot 10^{+84}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z - t}}\\ \mathbf{elif}\;a \leq 1.9705286202900668 \cdot 10^{-120}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;a \leq 3.690147293093053 \cdot 10^{-15}:\\ \;\;\;\;z \cdot \frac{y - x}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z - t}}\\ \end{array}\]

Alternative 9
Error	27.8
Cost	2509

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4933790013899865 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;y \leq 1.2830709588516494 \cdot 10^{-253}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 1.8832072673378197 \cdot 10^{-55}:\\ \;\;\;\;z \cdot \frac{y - x}{a - t}\\ \mathbf{elif}\;y \leq 1.0346537107323788 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 65946506011.980484:\\ \;\;\;\;z \cdot \frac{y - x}{a - t}\\ \mathbf{elif}\;y \leq 4.19826399328792 \cdot 10^{+69} \lor \neg \left(y \leq 8.736939133402569 \cdot 10^{+92}\right):\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 10
Error	27.8
Cost	2509

\[\begin{array}{l} \mathbf{if}\;y \leq -1.505204305228059 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;y \leq 3.347435105176827 \cdot 10^{-253}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 2.091129715433925 \cdot 10^{-55}:\\ \;\;\;\;z \cdot \frac{y - x}{a - t}\\ \mathbf{elif}\;y \leq 3.002885773472767 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 65946506011.980484:\\ \;\;\;\;z \cdot \frac{y - x}{a - t}\\ \mathbf{elif}\;y \leq 3.917320055467922 \cdot 10^{+69} \lor \neg \left(y \leq 8.736939133402569 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 11
Error	23.9
Cost	904

\[\begin{array}{l} \mathbf{if}\;a \leq -4.479083451051464 \cdot 10^{+84} \lor \neg \left(a \leq 7.461227417847208 \cdot 10^{-68}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \end{array}\]

Alternative 12
Error	30.3
Cost	1169

\[\begin{array}{l} \mathbf{if}\;y \leq -2.450248352175877 \cdot 10^{-21} \lor \neg \left(y \leq 9.417835694895733 \cdot 10^{-213} \lor \neg \left(y \leq 4.057792024377921 \cdot 10^{+69}\right) \land y \leq 8.736939133402569 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{y}{\frac{a - t}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 13
Error	36.0
Cost	706

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8190922429301292 \cdot 10^{+67}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 6.906028203678334 \cdot 10^{+85}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array}\]

Alternative 14
Error	45.8
Cost	64

\[x\]

Alternative 15
Error	61.8
Cost	64

\[1\]

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_18096 7.7

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

   4. Simplified 7.7

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

   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 0.8

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

   3. Simplified 0.8

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

   4. Simplified 0.8

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

4. Final simplification 7.2

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

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))))