Henrywood and Agarwal, Equation (12)

Average Error: 27.0 → 21.1

Time: 11.5s

Precision: binary64

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

↓

\[\begin{array}{l} t_0 := \sqrt{\sqrt{\frac{d}{h}}}\\ t_1 := 1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_4 := t_2 \cdot t_3\\ t_5 := t_4 \cdot t_1\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;t_4 \cdot \left(1 + \frac{h \cdot \left(D \cdot \frac{D}{\ell}\right)}{d \cdot \frac{\frac{d}{M}}{M}} \cdot -0.125\right)\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;t_1 \cdot \left(t_3 \cdot \left(t_0 \cdot t_0\right)\right)\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{+263}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2} \cdot -0.5}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt (/ d h))))
        (t_1 (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
        (t_2 (pow (/ d h) 0.5))
        (t_3 (pow (/ d l) 0.5))
        (t_4 (* t_2 t_3))
        (t_5 (* t_4 t_1)))
   (if (<= t_5 (- INFINITY))
     (* t_4 (+ 1.0 (* (/ (* h (* D (/ D l))) (* d (/ (/ d M) M))) -0.125)))
     (if (<= t_5 0.0)
       (* t_1 (* t_3 (* t_0 t_0)))
       (if (<= t_5 2e+263)
         (*
          t_2
          (*
           (sqrt (/ d l))
           (+ 1.0 (/ (* (pow (/ M (/ 2.0 (/ D d))) 2.0) -0.5) (/ l h)))))
         (* d (sqrt (/ 1.0 (* h l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(sqrt((d / h)));
	double t_1 = 1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l));
	double t_2 = pow((d / h), 0.5);
	double t_3 = pow((d / l), 0.5);
	double t_4 = t_2 * t_3;
	double t_5 = t_4 * t_1;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_4 * (1.0 + (((h * (D * (D / l))) / (d * ((d / M) / M))) * -0.125));
	} else if (t_5 <= 0.0) {
		tmp = t_1 * (t_3 * (t_0 * t_0));
	} else if (t_5 <= 2e+263) {
		tmp = t_2 * (sqrt((d / l)) * (1.0 + ((pow((M / (2.0 / (D / d))), 2.0) * -0.5) / (l / h))));
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(Math.sqrt((d / h)));
	double t_1 = 1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l));
	double t_2 = Math.pow((d / h), 0.5);
	double t_3 = Math.pow((d / l), 0.5);
	double t_4 = t_2 * t_3;
	double t_5 = t_4 * t_1;
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_4 * (1.0 + (((h * (D * (D / l))) / (d * ((d / M) / M))) * -0.125));
	} else if (t_5 <= 0.0) {
		tmp = t_1 * (t_3 * (t_0 * t_0));
	} else if (t_5 <= 2e+263) {
		tmp = t_2 * (Math.sqrt((d / l)) * (1.0 + ((Math.pow((M / (2.0 / (D / d))), 2.0) * -0.5) / (l / h))));
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

↓

def code(d, h, l, M, D):
	t_0 = math.sqrt(math.sqrt((d / h)))
	t_1 = 1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l))
	t_2 = math.pow((d / h), 0.5)
	t_3 = math.pow((d / l), 0.5)
	t_4 = t_2 * t_3
	t_5 = t_4 * t_1
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_4 * (1.0 + (((h * (D * (D / l))) / (d * ((d / M) / M))) * -0.125))
	elif t_5 <= 0.0:
		tmp = t_1 * (t_3 * (t_0 * t_0))
	elif t_5 <= 2e+263:
		tmp = t_2 * (math.sqrt((d / l)) * (1.0 + ((math.pow((M / (2.0 / (D / d))), 2.0) * -0.5) / (l / h))))
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = sqrt(sqrt(Float64(d / h)))
	t_1 = Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l)))
	t_2 = Float64(d / h) ^ 0.5
	t_3 = Float64(d / l) ^ 0.5
	t_4 = Float64(t_2 * t_3)
	t_5 = Float64(t_4 * t_1)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(t_4 * Float64(1.0 + Float64(Float64(Float64(h * Float64(D * Float64(D / l))) / Float64(d * Float64(Float64(d / M) / M))) * -0.125)));
	elseif (t_5 <= 0.0)
		tmp = Float64(t_1 * Float64(t_3 * Float64(t_0 * t_0)));
	elseif (t_5 <= 2e+263)
		tmp = Float64(t_2 * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64((Float64(M / Float64(2.0 / Float64(D / d))) ^ 2.0) * -0.5) / Float64(l / h)))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

↓

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(sqrt((d / h)));
	t_1 = 1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l));
	t_2 = (d / h) ^ 0.5;
	t_3 = (d / l) ^ 0.5;
	t_4 = t_2 * t_3;
	t_5 = t_4 * t_1;
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_4 * (1.0 + (((h * (D * (D / l))) / (d * ((d / M) / M))) * -0.125));
	elseif (t_5 <= 0.0)
		tmp = t_1 * (t_3 * (t_0 * t_0));
	elseif (t_5 <= 2e+263)
		tmp = t_2 * (sqrt((d / l)) * (1.0 + ((((M / (2.0 / (D / d))) ^ 2.0) * -0.5) / (l / h))));
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(t$95$4 * N[(1.0 + N[(N[(N[(h * N[(D * N[(D / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * N[(N[(d / M), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$1 * N[(t$95$3 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+263], N[(t$95$2 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[Power[N[(M / N[(2.0 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -inf.0

   1. Initial program 64.0

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 53.4

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   3. Taylor expanded in M around 0 59.4

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{\ell \cdot {d}^{2}}}\right) \]

   4. Simplified 50.9

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{0.125 \cdot \frac{h \cdot \left(D \cdot \frac{D}{\ell}\right)}{\frac{\frac{d}{M}}{M} \cdot d}}\right) \]

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -0.0

   1. Initial program 21.8

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 21.9

      \[\leadsto \left(\color{blue}{\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\frac{d}{h}}}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2.00000000000000003e263

   1. Initial program 0.9

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Applied egg-rr 1.1

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\left(0.5 \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}\right) \cdot h}{\ell}}\right) \]

   3. Applied egg-rr 1.3

      \[\leadsto \left(\color{blue}{\left({\left(\frac{d}{h}\right)}^{0.25} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}\right) \cdot h}{\ell}\right) \]

   4. Applied egg-rr 1.2

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - \frac{0.5 \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}}{\frac{\ell}{h}}\right)\right)\right)}^{1}} \]

   if 2.00000000000000003e263 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 62.6

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around inf 44.9

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 21.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 + \frac{h \cdot \left(D \cdot \frac{D}{\ell}\right)}{d \cdot \frac{\frac{d}{M}}{M}} \cdot -0.125\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot \left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\frac{d}{h}}}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+263}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.5} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{{\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2} \cdot -0.5}{\frac{\ell}{h}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022165 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))