Average Error: 14.1 → 0.2

Time: 17.1s

Precision: binary64

Cost: 21122

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -2.957723683841436 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 97778852.442119:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

\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 -2.957723683841436 \cdot 10^{+47}:\\
\;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\

\mathbf{elif}\;F \leq 97778852.442119:\\
\;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\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 -2.957723683841436e+47)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 97778852.442119)
     (-
      (/ 1.0 (/ (sin B) (* F (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5))))
      (/ x (tan B)))
     (- (/ 1.0 (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 <= -2.957723683841436e+47) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 97778852.442119) {
		tmp = (1.0 / (sin(B) / (F * pow(((x * 2.0) + (2.0 + (F * F))), -0.5)))) - (x / tan(B));
	} else {
		tmp = (1.0 / 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	50.6
Cost	72512

\[\left({\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5} \cdot \frac{\sqrt{F}}{\sqrt{\sin B}}\right) \cdot \left({\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5} \cdot \frac{\sqrt{F}}{\sqrt{\sin B}}\right) - \frac{x}{\tan B}\]

Alternative 2
Error	14.7
Cost	72064

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\tan B}}\]

Alternative 3
Error	14.3
Cost	67072

\[\sqrt[3]{\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \left(\sqrt[3]{\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \sqrt[3]{\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}\right) - \frac{x}{\tan B}\]

Alternative 4
Error	34.4
Cost	65536

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{\sqrt{x}}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}} \cdot \frac{\sqrt{x}}{\sqrt[3]{\tan B}}\]

Alternative 5
Error	50.6
Cost	59712

\[\left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25} \cdot \frac{\sqrt{F}}{\sqrt{\sin B}}\right) \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25} \cdot \frac{\sqrt{F}}{\sqrt{\sin B}}\right) - \frac{x}{\tan B}\]

Alternative 6
Error	39.4
Cost	59712

\[\left(\sqrt{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \sqrt{\frac{F}{\sin B}}\right) \cdot \left(\sqrt{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \sqrt{\frac{F}{\sin B}}\right) - \frac{x}{\tan B}\]

Alternative 7
Error	37.2
Cost	59136

\[\frac{\sqrt[3]{F} \cdot \sqrt[3]{F}}{\sqrt{\sin B}} \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{\sqrt[3]{F}}{\sqrt{\sin B}}\right) - \frac{x}{\tan B}\]

Alternative 8
Error	14.3
Cost	53760

\[\frac{F}{\sin B} \cdot \left(\sqrt[3]{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \left(\sqrt[3]{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \sqrt[3]{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}\right)\right) - \frac{x}{\tan B}\]

Alternative 9
Error	14.5
Cost	52864

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \sqrt[3]{\frac{x}{\tan B}} \cdot \left(\sqrt[3]{\frac{x}{\tan B}} \cdot \sqrt[3]{\frac{x}{\tan B}}\right)\]

Alternative 10
Error	14.6
Cost	52736

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{1}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}} \cdot \frac{x}{\sqrt[3]{\tan B}}\]

Alternative 11
Error	11.7
Cost	52736

\[\frac{1}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{F}{\sqrt[3]{\sin B}}\right) - \frac{x}{\tan B}\]

Alternative 12
Error	14.6
Cost	52608

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{\frac{x}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}}}{\sqrt[3]{\tan B}}\]

Alternative 13
Error	49.3
Cost	52608

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{\sqrt{x}}{\sqrt{\tan B}} \cdot \frac{\sqrt{x}}{\sqrt{\tan B}}\]

Alternative 14
Error	50.6
Cost	52608

\[\frac{\sqrt{F}}{\sqrt{\sin B}} \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{\sqrt{F}}{\sqrt{\sin B}}\right) - \frac{x}{\tan B}\]

Alternative 15
Error	14.3
Cost	47296

\[\frac{F}{\sin B} \cdot \left({\left(\sqrt[3]{2 \cdot x + \left(2 + F \cdot F\right)} \cdot \sqrt[3]{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5} \cdot {\left(\sqrt[3]{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5}\right) - \frac{x}{\tan B}\]

Alternative 16
Error	35.5
Cost	46912

\[\sqrt{\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \sqrt{\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} - \frac{x}{\tan B}\]

Alternative 17
Error	14.2
Cost	40256

\[\frac{F}{\sin B} \cdot \left(\sqrt{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}} \cdot \sqrt{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}\right) - \frac{x}{\tan B}\]

Alternative 18
Error	14.1
Cost	40256

\[\frac{F}{\sin B} \cdot \left({\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5} \cdot {\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5}\right) - \frac{x}{\tan B}\]

Alternative 19
Error	14.1
Cost	40256

\[{\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5} \cdot \left(\frac{F}{\sin B} \cdot {\left(\sqrt{2 \cdot x + \left(2 + F \cdot F\right)}\right)}^{-0.5}\right) - \frac{x}{\tan B}\]

Alternative 20
Error	38.8
Cost	39808

\[\left(\frac{1}{\sqrt{\sin B}} \cdot \frac{F}{\sqrt{\sin B}}\right) \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{\tan B}\]

Alternative 21
Error	14.5
Cost	39808

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{\tan B}\]

Alternative 22
Error	39.3
Cost	39808

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \sqrt{\frac{x}{\tan B}} \cdot \sqrt{\frac{x}{\tan B}}\]

Alternative 23
Error	33.0
Cost	39488

\[\log \left({\left(e^{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}\right)}^{\left(\frac{F}{\sin B}\right)}\right) - \frac{x}{\tan B}\]

Alternative 24
Error	34.1
Cost	33280

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \sqrt{x} \cdot \frac{\sqrt{x}}{\tan B}\]

Alternative 25
Error	37.8
Cost	33280

\[\sqrt{F} \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{\sqrt{F}}{\sin B}\right) - \frac{x}{\tan B}\]

Alternative 26
Error	22.6
Cost	33216

\[\sqrt[3]{{\left(\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}\right)}^{3}} - \frac{x}{\tan B}\]

Alternative 27
Error	16.6
Cost	33216

\[\frac{F}{\sin B} \cdot \sqrt[3]{{\left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}\right)}^{3}} - \frac{x}{\tan B}\]

Alternative 28
Error	34.1
Cost	33216

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \sqrt[3]{{\left(\frac{x}{\tan B}\right)}^{3}}\]

Alternative 29
Error	36.0
Cost	33152

\[e^{\log \left(\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}\right)} - \frac{x}{\tan B}\]

Alternative 30
Error	46.5
Cost	33152

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \log \left(e^{\frac{x}{\tan B}}\right)\]

Alternative 31
Error	40.5
Cost	33152

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - e^{\log \left(\frac{x}{\tan B}\right)}\]

Alternative 32
Error	14.1
Cost	27456

\[\frac{F}{\sin B} \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) - \frac{x}{\tan B}\]

Alternative 33
Error	14.1
Cost	27456

\[{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25} \cdot \left(\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) - \frac{x}{\tan B}\]

Alternative 34
Error	10.8
Cost	27456

\[\left(F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right) \cdot \frac{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}}{\sin B} - \frac{x}{\tan B}\]

Alternative 35
Error	10.8
Cost	27456

\[F \cdot \frac{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}}{\sin B} - \frac{x}{\tan B}\]

Alternative 36
Error	14.0
Cost	26880

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{\sin B} \cdot \cos B\]

Alternative 37
Error	10.8
Cost	26816

\[F \cdot \frac{{\left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.25}\right)}^{2}}{\sin B} - \frac{x}{\tan B}\]

Alternative 38
Error	14.1
Cost	20480

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - x \cdot \frac{1}{\tan B}\]

Alternative 39
Error	10.8
Cost	20480

\[\frac{1}{\sin B} \cdot \left(F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}\right) - \frac{x}{\tan B}\]

Alternative 40
Error	14.0
Cost	20480

\[{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B}\]

Alternative 41
Error	14.1
Cost	20480

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{1}{\frac{\tan B}{x}}\]

Alternative 42
Error	14.0
Cost	20480

\[{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}} - \frac{x}{\tan B}\]

Alternative 43
Error	10.8
Cost	20480

\[\frac{1}{\frac{\sin B}{F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\]

Alternative 44
Error	10.8
Cost	20480

\[F \cdot \left({\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B}\]

Alternative 45
Error	10.8
Cost	20352

\[F \cdot \frac{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\]

Alternative 46
Error	14.0
Cost	20352

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{\tan B}\]

Alternative 47
Error	10.7
Cost	20352

\[\frac{F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\]

Alternative 48
Error	30.1
Cost	20160

\[\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - \frac{x}{\tan B}\]

Alternative 49
Error	10.8
Cost	20160

\[\frac{F \cdot \sqrt{\frac{1}{2 + F \cdot F}}}{\sin B} - \frac{x}{\tan B}\]

Alternative 50
Error	14.0
Cost	20160

\[\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + F \cdot F}} - \frac{x}{\tan B}\]

Alternative 51
Error	30.0
Cost	20160

\[F \cdot \frac{\sqrt{\frac{1}{2 + 2 \cdot x}}}{\sin B} - \frac{x}{\tan B}\]

Alternative 52
Error	30.0
Cost	20160

\[\frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}}{\sin B} - \frac{x}{\tan B}\]

Alternative 53
Error	47.7
Cost	19968

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x\right)}^{-0.5} - \frac{x}{\tan B}\]

Alternative 54
Error	27.5
Cost	14400

\[F \cdot \left(\sqrt{\frac{1}{2 \cdot x + \left(2 + F \cdot F\right)}} \cdot \left(\frac{1}{B} + B \cdot 0.16666666666666666\right)\right) - \frac{x}{\tan B}\]

Alternative 55
Error	24.9
Cost	14016

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

Alternative 56
Error	33.0
Cost	13952

\[\frac{F}{\sin B} \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{B}\]

Alternative 57
Error	24.9
Cost	13952

\[{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\]

Alternative 58
Error	29.8
Cost	13952

\[F \cdot \frac{{\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\]

Alternative 59
Error	29.8
Cost	13952

\[\frac{F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\]

Alternative 60
Error	35.6
Cost	13504

\[\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{\tan B}\]

Alternative 61
Error	36.0
Cost	13504

\[\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{\tan B}\]

Alternative 62
Error	32.6
Cost	13504

\[F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{\tan B}\]

Alternative 63
Error	31.1
Cost	13248

\[\frac{-1}{\sin B} - \frac{x}{\tan B}\]

Alternative 64
Error	31.4
Cost	13248

\[\frac{1}{\sin B} - \frac{x}{\tan B}\]

Alternative 65
Error	60.9
Cost	64

\[1\]

Alternative 66
Error	62.2
Cost	64

\[0\]

Alternative 67
Error	60.9
Cost	64

\[-1\]

Error

Derivation

Split input into 3 regimes
if F < -2.9577236838414358e47
1. Initial program 28.7
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
2. Simplified28.6
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
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 -2.9577236838414358e47 < F < 97778852.442119002
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. Using strategy rm
7. Applied clear-num_binary640.3
  \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}}} - \frac{x}{\tan B}\]
8. Simplified0.3
  \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F \cdot {\left(2 \cdot x + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}}\]
if 97778852.442119002 < F
1. Initial program 25.6
  \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
2. Simplified25.6
  \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
3. Taylor expanded around inf 0.1
  \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B}\]
4. Simplified0.1
  \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.957723683841436 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 97778852.442119:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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 < -2.9577236838414358e47`

`if -2.9577236838414358e47 < F < 97778852.442119002`

`if 97778852.442119002 < F`

Reproduce