Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 15.4

Time: 14.6s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{\frac{\ell}{d}}}\\ t_1 := 1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.5}\\ t_3 := \sqrt{-d}\\ t_4 := 1 - {\left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d} \cdot \sqrt{0.5}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{t_3}{\sqrt{-h}} \cdot t_0\right) \cdot t_1\\ \mathbf{elif}\;d \leq 0:\\ \;\;\;\;\left(t_2 \cdot \frac{t_3}{\sqrt{-\ell}}\right) \cdot t_4\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+19}:\\ \;\;\;\;t_4 \cdot \left(t_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]

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 (/ 1.0 (sqrt (/ l d))))
        (t_1 (- 1.0 (/ (* h (* 0.5 (pow (* (/ M 2.0) (/ D d)) 2.0))) l)))
        (t_2 (pow (/ d h) 0.5))
        (t_3 (sqrt (- d)))
        (t_4
         (-
          1.0
          (pow (/ (* (/ (* M (* 0.5 D)) d) (sqrt 0.5)) (sqrt (/ l h))) 2.0))))
   (if (<= d -1.35e+52)
     (* (* (/ t_3 (sqrt (- h))) t_0) t_1)
     (if (<= d 0.0)
       (* (* t_2 (/ t_3 (sqrt (- l)))) t_4)
       (if (<= d 3.8e+19)
         (* t_4 (* t_2 (/ (sqrt d) (sqrt l))))
         (* t_1 (* t_0 (/ (sqrt d) (sqrt h)))))))))

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 / sqrt((l / d));
	double t_1 = 1.0 - ((h * (0.5 * pow(((M / 2.0) * (D / d)), 2.0))) / l);
	double t_2 = pow((d / h), 0.5);
	double t_3 = sqrt(-d);
	double t_4 = 1.0 - pow(((((M * (0.5 * D)) / d) * sqrt(0.5)) / sqrt((l / h))), 2.0);
	double tmp;
	if (d <= -1.35e+52) {
		tmp = ((t_3 / sqrt(-h)) * t_0) * t_1;
	} else if (d <= 0.0) {
		tmp = (t_2 * (t_3 / sqrt(-l))) * t_4;
	} else if (d <= 3.8e+19) {
		tmp = t_4 * (t_2 * (sqrt(d) / sqrt(l)));
	} else {
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	}
	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) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 / sqrt((l / d))
    t_1 = 1.0d0 - ((h * (0.5d0 * (((m / 2.0d0) * (d_1 / d)) ** 2.0d0))) / l)
    t_2 = (d / h) ** 0.5d0
    t_3 = sqrt(-d)
    t_4 = 1.0d0 - (((((m * (0.5d0 * d_1)) / d) * sqrt(0.5d0)) / sqrt((l / h))) ** 2.0d0)
    if (d <= (-1.35d+52)) then
        tmp = ((t_3 / sqrt(-h)) * t_0) * t_1
    else if (d <= 0.0d0) then
        tmp = (t_2 * (t_3 / sqrt(-l))) * t_4
    else if (d <= 3.8d+19) then
        tmp = t_4 * (t_2 * (sqrt(d) / sqrt(l)))
    else
        tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)))
    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.sqrt((l / d));
	double t_1 = 1.0 - ((h * (0.5 * Math.pow(((M / 2.0) * (D / d)), 2.0))) / l);
	double t_2 = Math.pow((d / h), 0.5);
	double t_3 = Math.sqrt(-d);
	double t_4 = 1.0 - Math.pow(((((M * (0.5 * D)) / d) * Math.sqrt(0.5)) / Math.sqrt((l / h))), 2.0);
	double tmp;
	if (d <= -1.35e+52) {
		tmp = ((t_3 / Math.sqrt(-h)) * t_0) * t_1;
	} else if (d <= 0.0) {
		tmp = (t_2 * (t_3 / Math.sqrt(-l))) * t_4;
	} else if (d <= 3.8e+19) {
		tmp = t_4 * (t_2 * (Math.sqrt(d) / Math.sqrt(l)));
	} else {
		tmp = t_1 * (t_0 * (Math.sqrt(d) / Math.sqrt(h)));
	}
	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.sqrt((l / d))
	t_1 = 1.0 - ((h * (0.5 * math.pow(((M / 2.0) * (D / d)), 2.0))) / l)
	t_2 = math.pow((d / h), 0.5)
	t_3 = math.sqrt(-d)
	t_4 = 1.0 - math.pow(((((M * (0.5 * D)) / d) * math.sqrt(0.5)) / math.sqrt((l / h))), 2.0)
	tmp = 0
	if d <= -1.35e+52:
		tmp = ((t_3 / math.sqrt(-h)) * t_0) * t_1
	elif d <= 0.0:
		tmp = (t_2 * (t_3 / math.sqrt(-l))) * t_4
	elif d <= 3.8e+19:
		tmp = t_4 * (t_2 * (math.sqrt(d) / math.sqrt(l)))
	else:
		tmp = t_1 * (t_0 * (math.sqrt(d) / math.sqrt(h)))
	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 / sqrt(Float64(l / 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(d / h) ^ 0.5
	t_3 = sqrt(Float64(-d))
	t_4 = Float64(1.0 - (Float64(Float64(Float64(Float64(M * Float64(0.5 * D)) / d) * sqrt(0.5)) / sqrt(Float64(l / h))) ^ 2.0))
	tmp = 0.0
	if (d <= -1.35e+52)
		tmp = Float64(Float64(Float64(t_3 / sqrt(Float64(-h))) * t_0) * t_1);
	elseif (d <= 0.0)
		tmp = Float64(Float64(t_2 * Float64(t_3 / sqrt(Float64(-l)))) * t_4);
	elseif (d <= 3.8e+19)
		tmp = Float64(t_4 * Float64(t_2 * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(t_1 * Float64(t_0 * Float64(sqrt(d) / sqrt(h))));
	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((l / d));
	t_1 = 1.0 - ((h * (0.5 * (((M / 2.0) * (D / d)) ^ 2.0))) / l);
	t_2 = (d / h) ^ 0.5;
	t_3 = sqrt(-d);
	t_4 = 1.0 - (((((M * (0.5 * D)) / d) * sqrt(0.5)) / sqrt((l / h))) ^ 2.0);
	tmp = 0.0;
	if (d <= -1.35e+52)
		tmp = ((t_3 / sqrt(-h)) * t_0) * t_1;
	elseif (d <= 0.0)
		tmp = (t_2 * (t_3 / sqrt(-l))) * t_4;
	elseif (d <= 3.8e+19)
		tmp = t_4 * (t_2 * (sqrt(d) / sqrt(l)));
	else
		tmp = t_1 * (t_0 * (sqrt(d) / sqrt(h)));
	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[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $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[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[Power[N[(N[(N[(N[(M * N[(0.5 * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.35e+52], N[(N[(N[(t$95$3 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[d, 0.0], N[(N[(t$95$2 * N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[d, 3.8e+19], N[(t$95$4 * N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t$95$0 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $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 < -1.35e52

   1. Initial program 25.8

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

   2. Applied egg-rr 24.7

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

   3. Applied egg-rr 24.2

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

   4. Applied egg-rr 11.8

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

   if -1.35e52 < d < 0.0

   1. Initial program 26.3

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

   2. Applied egg-rr 26.0

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

   3. Applied egg-rr 24.0

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

   4. Applied egg-rr 18.1

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

   if 0.0 < d < 3.8e19

   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 28.7

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

   3. Applied egg-rr 26.0

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

   4. Applied egg-rr 19.7

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

   if 3.8e19 < d

   1. Initial program 24.4

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

   2. Applied egg-rr 23.3

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

   3. Applied egg-rr 22.9

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

   4. Applied egg-rr 10.8

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

4. Final simplification 15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+52}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 0:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - {\left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d} \cdot \sqrt{0.5}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+19}:\\ \;\;\;\;\left(1 - {\left(\frac{\frac{M \cdot \left(0.5 \cdot D\right)}{d} \cdot \sqrt{0.5}}{\sqrt{\frac{\ell}{h}}}\right)}^{2}\right) \cdot \left({\left(\frac{d}{h}\right)}^{0.5} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)}{\ell}\right) \cdot \left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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)))))