VandenBroeck and Keller, Equation (23)

Average Error: 14.7 → 0.2

Time: 15.2s

Precision: binary64

Cost: 20994

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -7.377918106249466 \cdot 10^{+84}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 94181933.04629011:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

FPCore

(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 -7.377918106249466e+84)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 94181933.04629011)
     (-
      (/ (* F (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (sin B))
      (/ x (tan B)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

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 <= -7.377918106249466e+84) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 94181933.04629011) {
		tmp = ((F * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) / sin(B)) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Alternatives

Alternative 1
Error	0.2
Cost	20802

\[\begin{array}{l} \mathbf{if}\;F \leq -1.74275693971342 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 122347326.68899404:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 2
Error	0.6
Cost	20802

\[\begin{array}{l} \mathbf{if}\;F \leq -1.8011926095838173 \cdot 10^{+84}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 473578177.1459986:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 3
Error	5.5
Cost	14915

\[\begin{array}{l} \mathbf{if}\;F \leq -0.03224115561723573:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7.764958837944096 \cdot 10^{-109}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4530646468251453 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 4
Error	10.6
Cost	14594

\[\begin{array}{l} \mathbf{if}\;F \leq -0.0008089251371989151:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.2181432083136693 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 5
Error	19.4
Cost	13569

\[\begin{array}{l} \mathbf{if}\;F \leq -4.225797020350178 \cdot 10^{-301}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Alternative 6
Error	30.5
Cost	13248

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

Alternative 7
Error	60.9
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if F < -7.37791810624946594e84

   1. Initial program 33.7

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

   2. Simplified 33.6

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

   3. Taylor expanded around -inf 0.1

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

   4. Simplified 0.1

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

   if -7.37791810624946594e84 < F < 94181933.0462901145

   1. Initial program 1.0

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}}\]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 0.3

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

   5. Simplified 0.3

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

   6. Simplified 0.3

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

   if 94181933.0462901145 < F

   1. Initial program 25.8

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

   2. Simplified 25.7

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

   3. Taylor expanded around inf 0.2

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

   4. Simplified 0.2

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.377918106249466 \cdot 10^{+84}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 94181933.04629011:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))))