Henrywood and Agarwal, Equation (12)

Average Error: 27.0 → 19.2

Time: 13.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_2 := \sqrt{-d}\\ t_3 := \left(\frac{t_2}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_0\\ \mathbf{if}\;d \leq -2.2077046003697104 \cdot 10^{-20}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -1.0053658497468265 \cdot 10^{-230}:\\ \;\;\;\;\left(t_1 \cdot \frac{t_2}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 5.16639765631 \cdot 10^{-310}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 6.241184318453877 \cdot 10^{+90}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

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
         (-
          1.0
          (pow (* (sqrt (/ h l)) (* (* (/ M 2.0) (/ D d)) (sqrt 0.5))) 2.0)))
        (t_1 (pow (/ d h) 0.5))
        (t_2 (sqrt (- d)))
        (t_3 (* (* (/ t_2 (sqrt (- h))) (pow (/ d l) 0.5)) t_0)))
   (if (<= d -2.2077046003697104e-20)
     t_3
     (if (<= d -1.0053658497468265e-230)
       (*
        (* t_1 (/ t_2 (sqrt (- l))))
        (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))))
       (if (<= d 5.16639765631e-310)
         t_3
         (if (<= d 6.241184318453877e+90)
           (* t_0 (* t_1 (/ (sqrt d) (sqrt l))))
           (* 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 = 1.0 - pow((sqrt((h / l)) * (((M / 2.0) * (D / d)) * sqrt(0.5))), 2.0);
	double t_1 = pow((d / h), 0.5);
	double t_2 = sqrt(-d);
	double t_3 = ((t_2 / sqrt(-h)) * pow((d / l), 0.5)) * t_0;
	double tmp;
	if (d <= -2.2077046003697104e-20) {
		tmp = t_3;
	} else if (d <= -1.0053658497468265e-230) {
		tmp = (t_1 * (t_2 / sqrt(-l))) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	} else if (d <= 5.16639765631e-310) {
		tmp = t_3;
	} else if (d <= 6.241184318453877e+90) {
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = d * sqrt((1.0 / (h * 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) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 - ((sqrt((h / l)) * (((m / 2.0d0) * (d_1 / d)) * sqrt(0.5d0))) ** 2.0d0)
    t_1 = (d / h) ** 0.5d0
    t_2 = sqrt(-d)
    t_3 = ((t_2 / sqrt(-h)) * ((d / l) ** 0.5d0)) * t_0
    if (d <= (-2.2077046003697104d-20)) then
        tmp = t_3
    else if (d <= (-1.0053658497468265d-230)) then
        tmp = (t_1 * (t_2 / sqrt(-l))) * (1.0d0 - ((h / l) * (0.5d0 * (((m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    else if (d <= 5.16639765631d-310) then
        tmp = t_3
    else if (d <= 6.241184318453877d+90) then
        tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)))
    else
        tmp = d * sqrt((1.0d0 / (h * 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 = 1.0 - Math.pow((Math.sqrt((h / l)) * (((M / 2.0) * (D / d)) * Math.sqrt(0.5))), 2.0);
	double t_1 = Math.pow((d / h), 0.5);
	double t_2 = Math.sqrt(-d);
	double t_3 = ((t_2 / Math.sqrt(-h)) * Math.pow((d / l), 0.5)) * t_0;
	double tmp;
	if (d <= -2.2077046003697104e-20) {
		tmp = t_3;
	} else if (d <= -1.0053658497468265e-230) {
		tmp = (t_1 * (t_2 / Math.sqrt(-l))) * (1.0 - ((h / l) * (0.5 * Math.pow(((M * D) / (d * 2.0)), 2.0))));
	} else if (d <= 5.16639765631e-310) {
		tmp = t_3;
	} else if (d <= 6.241184318453877e+90) {
		tmp = t_0 * (t_1 * (Math.sqrt(d) / Math.sqrt(l)));
	} 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 = 1.0 - math.pow((math.sqrt((h / l)) * (((M / 2.0) * (D / d)) * math.sqrt(0.5))), 2.0)
	t_1 = math.pow((d / h), 0.5)
	t_2 = math.sqrt(-d)
	t_3 = ((t_2 / math.sqrt(-h)) * math.pow((d / l), 0.5)) * t_0
	tmp = 0
	if d <= -2.2077046003697104e-20:
		tmp = t_3
	elif d <= -1.0053658497468265e-230:
		tmp = (t_1 * (t_2 / math.sqrt(-l))) * (1.0 - ((h / l) * (0.5 * math.pow(((M * D) / (d * 2.0)), 2.0))))
	elif d <= 5.16639765631e-310:
		tmp = t_3
	elif d <= 6.241184318453877e+90:
		tmp = t_0 * (t_1 * (math.sqrt(d) / math.sqrt(l)))
	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 = Float64(1.0 - (Float64(sqrt(Float64(h / l)) * Float64(Float64(Float64(M / 2.0) * Float64(D / d)) * sqrt(0.5))) ^ 2.0))
	t_1 = Float64(d / h) ^ 0.5
	t_2 = sqrt(Float64(-d))
	t_3 = Float64(Float64(Float64(t_2 / sqrt(Float64(-h))) * (Float64(d / l) ^ 0.5)) * t_0)
	tmp = 0.0
	if (d <= -2.2077046003697104e-20)
		tmp = t_3;
	elseif (d <= -1.0053658497468265e-230)
		tmp = Float64(Float64(t_1 * Float64(t_2 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))));
	elseif (d <= 5.16639765631e-310)
		tmp = t_3;
	elseif (d <= 6.241184318453877e+90)
		tmp = Float64(t_0 * Float64(t_1 * Float64(sqrt(d) / sqrt(l))));
	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 = 1.0 - ((sqrt((h / l)) * (((M / 2.0) * (D / d)) * sqrt(0.5))) ^ 2.0);
	t_1 = (d / h) ^ 0.5;
	t_2 = sqrt(-d);
	t_3 = ((t_2 / sqrt(-h)) * ((d / l) ^ 0.5)) * t_0;
	tmp = 0.0;
	if (d <= -2.2077046003697104e-20)
		tmp = t_3;
	elseif (d <= -1.0053658497468265e-230)
		tmp = (t_1 * (t_2 / sqrt(-l))) * (1.0 - ((h / l) * (0.5 * (((M * D) / (d * 2.0)) ^ 2.0))));
	elseif (d <= 5.16639765631e-310)
		tmp = t_3;
	elseif (d <= 6.241184318453877e+90)
		tmp = t_0 * (t_1 * (sqrt(d) / sqrt(l)));
	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[(1.0 - N[Power[N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[d, -2.2077046003697104e-20], t$95$3, If[LessEqual[d, -1.0053658497468265e-230], N[(N[(t$95$1 * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.16639765631e-310], t$95$3, If[LessEqual[d, 6.241184318453877e+90], N[(t$95$0 * N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $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 d < -2.2077046003697104e-20 or -1.00536584974682647e-230 < d < 5.166397656309988e-310

   1. Initial program 26.2

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

   2. Applied egg-rr 25.4

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

   3. Applied egg-rr 16.5

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

   if -2.2077046003697104e-20 < d < -1.00536584974682647e-230

   1. Initial program 28.9

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

   2. Applied egg-rr 22.8

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

   if 5.166397656309988e-310 < d < 6.2411843184538766e90

   1. Initial program 26.8

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

   2. Applied egg-rr 25.7

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

   3. Applied egg-rr 20.8

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

   if 6.2411843184538766e90 < d

   1. Initial program 26.9

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

   2. Applied egg-rr 25.7

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

   3. Taylor expanded in d around inf 17.8

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

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2077046003697104 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq -1.0053658497468265 \cdot 10^{-230}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 5.16639765631 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 6.241184318453877 \cdot 10^{+90}:\\ \;\;\;\;\left(1 - {\left(\sqrt{\frac{h}{\ell}} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right)\right)}^{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

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)))))