VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 0.2

Time: 10.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sin B}{F}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 15000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{t_0} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot \frac{x}{F} + \left(\sin B + \frac{\sin B}{F \cdot F}\right)} - t_1\\ \end{array} \]

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
 (let* ((t_0 (/ (sin B) F)) (t_1 (/ x (tan B))))
   (if (<= F -650000.0)
     (- (/ -1.0 (sin B)) (* (/ x (sin B)) (cos B)))
     (if (<= F 15000.0)
       (- (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) t_0) t_1)
       (- (/ 1.0 (+ (* t_0 (/ x F)) (+ (sin B) (/ (sin B) (* F F))))) t_1)))))

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 t_0 = sin(B) / F;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -650000.0) {
		tmp = (-1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	} else if (F <= 15000.0) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / t_0) - t_1;
	} else {
		tmp = (1.0 / ((t_0 * (x / F)) + (sin(B) + (sin(B) / (F * F))))) - t_1;
	}
	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)
	t_0 = Float64(sin(B) / F)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -650000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	elseif (F <= 15000.0)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / t_0) - t_1);
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(t_0 * Float64(x / F)) + Float64(sin(B) + Float64(sin(B) / Float64(F * F))))) - t_1);
	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_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -650000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 15000.0], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(1.0 / N[(N[(t$95$0 * N[(x / F), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[B], $MachinePrecision] + N[(N[Sin[B], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -6.5e5

   1. Initial program 25.5

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

   2. Simplified 19.3

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

   3. Taylor expanded in x around 0 19.3

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

   4. Simplified 19.3

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

   5. Taylor expanded in F around -inf 0.2

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

   if -6.5e5 < F < 15000

   1. Initial program 0.4

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

   2. Simplified 0.3

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

   3. Applied egg-rr 0.3

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

   4. Applied egg-rr 0.3

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

   if 15000 < F

   1. Initial program 25.5

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

   2. Simplified 19.5

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

   3. Applied egg-rr 19.5

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

   4. Taylor expanded in F around inf 0.1

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

   5. Simplified 0.1

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin B}{F} \cdot \frac{x}{F} + \left(\sin B + \frac{\sin B}{F \cdot F}\right)}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;F \leq 15000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F} \cdot \frac{x}{F} + \left(\sin B + \frac{\sin B}{F \cdot F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))))