Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 18.1

Time: 13.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := 1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\\ t_2 := \frac{1}{\sqrt{\frac{h}{d}}}\\ \mathbf{if}\;h \leq -5.088748267859237 \cdot 10^{-188}:\\ \;\;\;\;\left(t_2 \cdot \frac{t_0}{\sqrt{-\ell}}\right) \cdot t_1\\ \mathbf{elif}\;h \leq 1.76573110744144 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t_0}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 5.057362700240132 \cdot 10^{-19}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

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 (- d)))
        (t_1 (- 1.0 (/ (* h (* 0.5 (pow (* (/ M 2.0) (/ D d)) 2.0))) l)))
        (t_2 (/ 1.0 (sqrt (/ h d)))))
   (if (<= h -5.088748267859237e-188)
     (* (* t_2 (/ t_0 (sqrt (- l)))) t_1)
     (if (<= h 1.76573110744144e-310)
       (*
        (* (/ t_0 (sqrt (- h))) (pow (/ d l) 0.5))
        (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
       (if (<= h 5.057362700240132e-19)
         (* d (sqrt (/ 1.0 (* h l))))
         (* t_1 (* t_2 (/ (sqrt d) (sqrt 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(-d);
	double t_1 = 1.0 - ((h * (0.5 * pow(((M / 2.0) * (D / d)), 2.0))) / l);
	double t_2 = 1.0 / sqrt((h / d));
	double tmp;
	if (h <= -5.088748267859237e-188) {
		tmp = (t_2 * (t_0 / sqrt(-l))) * t_1;
	} else if (h <= 1.76573110744144e-310) {
		tmp = ((t_0 / sqrt(-h)) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	} else if (h <= 5.057362700240132e-19) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else {
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

↓

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = 1.0d0 - ((h * (0.5d0 * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0))) / l)
    t_2 = 1.0d0 / sqrt((h / d))
    if (h <= (-5.088748267859237d-188)) then
        tmp = (t_2 * (t_0 / sqrt(-l))) * t_1
    else if (h <= 1.76573110744144d-310) then
        tmp = ((t_0 / sqrt(-h)) * ((d / l) ** 0.5d0)) * (1.0d0 - ((0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0)) * (h / l)))
    else if (h <= 5.057362700240132d-19) then
        tmp = d * sqrt((1.0d0 / (h * l)))
    else
        tmp = t_1 * (t_2 * (sqrt(d) / sqrt(l)))
    end if
    code = tmp
end function

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(-d);
	double t_1 = 1.0 - ((h * (0.5 * Math.pow(((M / 2.0) * (D / d)), 2.0))) / l);
	double t_2 = 1.0 / Math.sqrt((h / d));
	double tmp;
	if (h <= -5.088748267859237e-188) {
		tmp = (t_2 * (t_0 / Math.sqrt(-l))) * t_1;
	} else if (h <= 1.76573110744144e-310) {
		tmp = ((t_0 / Math.sqrt(-h)) * Math.pow((d / l), 0.5)) * (1.0 - ((0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	} else if (h <= 5.057362700240132e-19) {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = t_1 * (t_2 * (Math.sqrt(d) / Math.sqrt(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(-d)
	t_1 = 1.0 - ((h * (0.5 * math.pow(((M / 2.0) * (D / d)), 2.0))) / l)
	t_2 = 1.0 / math.sqrt((h / d))
	tmp = 0
	if h <= -5.088748267859237e-188:
		tmp = (t_2 * (t_0 / math.sqrt(-l))) * t_1
	elif h <= 1.76573110744144e-310:
		tmp = ((t_0 / math.sqrt(-h)) * math.pow((d / l), 0.5)) * (1.0 - ((0.5 * math.pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))
	elif h <= 5.057362700240132e-19:
		tmp = d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = t_1 * (t_2 * (math.sqrt(d) / math.sqrt(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(Float64(-d))
	t_1 = Float64(1.0 - Float64(Float64(h * Float64(0.5 * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0))) / l))
	t_2 = Float64(1.0 / sqrt(Float64(h / d)))
	tmp = 0.0
	if (h <= -5.088748267859237e-188)
		tmp = Float64(Float64(t_2 * Float64(t_0 / sqrt(Float64(-l)))) * t_1);
	elseif (h <= 1.76573110744144e-310)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))));
	elseif (h <= 5.057362700240132e-19)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(t_1 * Float64(t_2 * Float64(sqrt(d) / sqrt(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(-d);
	t_1 = 1.0 - ((h * (0.5 * (((M / 2.0) * (D / d)) ^ 2.0))) / l);
	t_2 = 1.0 / sqrt((h / d));
	tmp = 0.0;
	if (h <= -5.088748267859237e-188)
		tmp = (t_2 * (t_0 / sqrt(-l))) * t_1;
	elseif (h <= 1.76573110744144e-310)
		tmp = ((t_0 / sqrt(-h)) * ((d / l) ^ 0.5)) * (1.0 - ((0.5 * (((M * D) / (d * 2.0)) ^ 2.0)) * (h / l)));
	elseif (h <= 5.057362700240132e-19)
		tmp = d * sqrt((1.0 / (h * l)));
	else
		tmp = t_1 * (t_2 * (sqrt(d) / sqrt(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[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(h * N[(0.5 * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5.088748267859237e-188], N[(N[(t$95$2 * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[h, 1.76573110744144e-310], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], If[LessEqual[h, 5.057362700240132e-19], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[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 h < -5.0887482678592368e-188

   1. Initial program 24.1

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

   2. Applied egg-rr 23.9

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

   3. Applied egg-rr 23.2

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

   4. Applied egg-rr 17.0

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

   if -5.0887482678592368e-188 < h < 1.76573110744144e-310

   1. Initial program 35.6

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

   2. Applied egg-rr 17.4

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

   if 1.76573110744144e-310 < h < 5.0573627002401319e-19

   1. Initial program 28.4

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

   2. Taylor expanded in d around inf 21.2

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   if 5.0573627002401319e-19 < h

   1. Initial program 25.0

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

   2. Applied egg-rr 25.3

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

   3. Applied egg-rr 24.3

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

   4. Applied egg-rr 17.2

      \[\leadsto \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right) \cdot h}{\ell}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5.088748267859237 \cdot 10^{-188}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;h \leq 1.76573110744144 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;h \leq 5.057362700240132 \cdot 10^{-19}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right) \cdot \left(\frac{1}{\sqrt{\frac{h}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))