Average Error: 13.7 → 0.2

Time: 4.3min

Precision: binary64

Cost: 20994

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -2.2682253394869427 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 254769.1791808347:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}

↓

\begin{array}{l}
\mathbf{if}\;F \leq -2.2682253394869427 \cdot 10^{+36}:\\
\;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 254769.1791808347:\\
\;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\

\end{array}

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

↓

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.2682253394869427e+36)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 254769.1791808347)
     (-
      (/ (* F (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (sin B))
      (/ x (tan B)))
     (- (/ (- 1.0 (+ (/ x (* F F)) (/ 1.0 (* F F)))) (sin B)) (/ x (tan B))))))

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

↓

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2682253394869427e+36) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 254769.1791808347) {
		tmp = ((F * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) / sin(B)) - (x / tan(B));
	} else {
		tmp = ((1.0 - ((x / (F * F)) + (1.0 / (F * F)))) / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `F`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.2
Cost	20802

\[\begin{array}{l} \mathbf{if}\;F \leq -2.1390484231469626 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 228986.99893373696:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 2
Error	0.5
Cost	20802

\[\begin{array}{l} \mathbf{if}\;F \leq -1.435136797446235:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 26.53211519579416:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 3
Error	5.4
Cost	14979

\[\begin{array}{l} \mathbf{if}\;F \leq -73.36809345229936:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2402944923380242 \cdot 10^{-127}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 283.8050133319312:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 4
Error	5.4
Cost	14915

\[\begin{array}{l} \mathbf{if}\;F \leq -75.80811335455446:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2402944923380242 \cdot 10^{-127}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6486.398982731933:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 5
Error	7.0
Cost	15236

\[\begin{array}{l} \mathbf{if}\;F \leq -75.80811335455446:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2008151000659857 \cdot 10^{-127}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.080216094282791 \cdot 10^{-187}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 541788.5644028311:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 6
Error	7.0
Cost	15236

\[\begin{array}{l} \mathbf{if}\;F \leq -75.80811335455446:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2402944923380242 \cdot 10^{-127}:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.3863620014432923 \cdot 10^{-180}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 987277.5500405453:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 7
Error	10.9
Cost	13890

\[\begin{array}{l} \mathbf{if}\;F \leq -6.95601292148594 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.438733885708423 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 8
Error	21.5
Cost	13569

\[\begin{array}{l} \mathbf{if}\;F \leq -2.1242530184207155 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \end{array}\]

Alternative 9
Error	26.6
Cost	13826

\[\begin{array}{l} \mathbf{if}\;F \leq -9.525976295326883 \cdot 10^{+91}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.399891388730018 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \end{array}\]

Alternative 10
Error	31.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7656457365202159 \cdot 10^{-12} \lor \neg \left(x \leq 1.6959097733192 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array}\]

Alternative 11
Error	42.6
Cost	6913

\[\begin{array}{l} \mathbf{if}\;F \leq -6735.992858857284:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array}\]

Alternative 12
Error	44.9
Cost	641

\[\begin{array}{l} \mathbf{if}\;F \leq -3.0404691183763064 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array}\]

Alternative 13
Error	48.7
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -3.689161297552655 \cdot 10^{-115} \lor \neg \left(x \leq -8.814364352337864 \cdot 10^{-157} \lor \neg \left(x \leq -1.1318218994911311 \cdot 10^{-242}\right) \land x \leq 2.3385530704709698 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array}\]

Alternative 14
Error	57.3
Cost	192

\[\frac{-1}{B}\]

Alternative 15
Error	60.8
Cost	64

\[1\]

Derivation

Split input into 3 regimes
if F < -2.2682253394869427e36
1. Initial program 26.6
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
2. Simplified26.6
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
3. Taylor expanded around -inf 0.2
  \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B}\]
4. Simplified0.2
  \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}}\]
if -2.2682253394869427e36 < F < 254769.179180834704
1. Initial program 0.5
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
2. Simplified0.4
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
3. Using strategy rm
4. Applied associate-*l/_binary640.3
  \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B}\]
5. Simplified0.3
  \[\leadsto \frac{\color{blue}{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B}\]
6. Simplified0.3
  \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}}\]
if 254769.179180834704 < F
1. Initial program 25.0
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
2. Simplified24.9
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
3. Using strategy rm
4. Applied associate-*l/_binary6419.2
  \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B}\]
5. Simplified19.2
  \[\leadsto \frac{\color{blue}{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B}\]
6. Taylor expanded around inf 0.2
  \[\leadsto \frac{\color{blue}{1 - \left(\frac{x}{{F}^{2}} + \frac{1}{{F}^{2}}\right)}}{\sin B} - \frac{x}{\tan B}\]
7. Simplified0.2
  \[\leadsto \frac{\color{blue}{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B}\]
8. Simplified0.2
  \[\leadsto \color{blue}{\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}}\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2682253394869427 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 254769.1791808347:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

Error

Try it out

Alternatives

Error

Derivation

`if F < -2.2682253394869427e36`

`if -2.2682253394869427e36 < F < 254769.179180834704`

`if 254769.179180834704 < F`

Reproduce