VandenBroeck and Keller, Equation (23)

Average Error: 13.7 → 0.6

Time: 11.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -1.5229173632385416 \cdot 10^{+91}:\\ \;\;\;\;\frac{-1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

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 -1.5229173632385416e+91)
   (- (/ -1.0 (sin B)) (pow (/ (tan B) x) -1.0))
   (if (<= F 9.5e-8)
     (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) (/ x (tan B)))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin 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 <= -1.5229173632385416e+91) {
		tmp = (-1.0 / sin(B)) - pow((tan(B) / x), -1.0);
	} else if (F <= 9.5e-8) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

↓

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5229173632385416e+91)
		tmp = Float64(Float64(-1.0 / sin(B)) - (Float64(tan(B) / x) ^ -1.0));
	elseif (F <= 9.5e-8)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[F_, B_, x_] := If[LessEqual[F, -1.5229173632385416e+91], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[Power[N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-8], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5229173632385416e91

   1. Initial program 32.1

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

   2. Simplified 26.5

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

   3. Applied egg-rr 26.6

      \[\leadsto F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \color{blue}{{\left(\frac{\tan B}{x}\right)}^{-1}} \]

   4. Taylor expanded in F around -inf 0.2

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - {\left(\frac{\tan B}{x}\right)}^{-1} \]

   if -1.5229173632385416e91 < F < 9.50000000000000036e-8

   1. Initial program 0.9

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

   2. Simplified 0.3

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

   if 9.50000000000000036e-8 < F

   1. Initial program 24.1

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

   2. Simplified 17.9

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

   3. Taylor expanded in x around 0 17.9

      \[\leadsto F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \color{blue}{\frac{\cos B \cdot x}{\sin B}} \]

   4. Taylor expanded in F around inf 1.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5229173632385416 \cdot 10^{+91}:\\ \;\;\;\;\frac{-1}{\sin B} - {\left(\frac{\tan B}{x}\right)}^{-1}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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))))))