Henrywood and Agarwal, Equation (9a)

Average Error: 14.4 → 9.7

Time: 16.7s

Precision: binary64

Cost: 20932

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

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

↓

\[\begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;h \leq 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(0.5 \cdot t_0\right) \cdot \sqrt{h}}{\ell} \cdot \left(\sqrt{h} \cdot \left(t_0 \cdot -0.5\right)\right)}\\ \end{array} \]

FPCore

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))

↓

(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= h 1e-307)
     (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* M 0.5) (/ D d)) 2.0)) l))))
     (*
      w0
      (sqrt
       (+
        1.0
        (* (/ (* (* 0.5 t_0) (sqrt h)) l) (* (sqrt h) (* t_0 -0.5)))))))))

C

double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / d);
	double tmp;
	if (h <= 1e-307) {
		tmp = w0 * sqrt((1.0 - ((h * pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else {
		tmp = w0 * sqrt((1.0 + ((((0.5 * t_0) * sqrt(h)) / l) * (sqrt(h) * (t_0 * -0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function

↓

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d / d_1)
    if (h <= 1d-307) then
        tmp = w0 * sqrt((1.0d0 - ((h * (((m * 0.5d0) * (d / d_1)) ** 2.0d0)) / l)))
    else
        tmp = w0 * sqrt((1.0d0 + ((((0.5d0 * t_0) * sqrt(h)) / l) * (sqrt(h) * (t_0 * (-0.5d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}

↓

public static double code(double w0, double M, double D, double h, double l, double d) {
	double t_0 = M * (D / d);
	double tmp;
	if (h <= 1e-307) {
		tmp = w0 * Math.sqrt((1.0 - ((h * Math.pow(((M * 0.5) * (D / d)), 2.0)) / l)));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((((0.5 * t_0) * Math.sqrt(h)) / l) * (Math.sqrt(h) * (t_0 * -0.5)))));
	}
	return tmp;
}

Python

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))

↓

def code(w0, M, D, h, l, d):
	t_0 = M * (D / d)
	tmp = 0
	if h <= 1e-307:
		tmp = w0 * math.sqrt((1.0 - ((h * math.pow(((M * 0.5) * (D / d)), 2.0)) / l)))
	else:
		tmp = w0 * math.sqrt((1.0 + ((((0.5 * t_0) * math.sqrt(h)) / l) * (math.sqrt(h) * (t_0 * -0.5)))))
	return tmp

Julia

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end

↓

function code(w0, M, D, h, l, d)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (h <= 1e-307)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h * (Float64(Float64(M * 0.5) * Float64(D / d)) ^ 2.0)) / l))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(0.5 * t_0) * sqrt(h)) / l) * Float64(sqrt(h) * Float64(t_0 * -0.5))))));
	end
	return tmp
end

MATLAB

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end

↓

function tmp_2 = code(w0, M, D, h, l, d)
	t_0 = M * (D / d);
	tmp = 0.0;
	if (h <= 1e-307)
		tmp = w0 * sqrt((1.0 - ((h * (((M * 0.5) * (D / d)) ^ 2.0)) / l)));
	else
		tmp = w0 * sqrt((1.0 + ((((0.5 * t_0) * sqrt(h)) / l) * (sqrt(h) * (t_0 * -0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[w0_, M_, D_, h_, l_, d_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, 1e-307], N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h * N[Power[N[(N[(M * 0.5), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(0.5 * t$95$0), $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[Sqrt[h], $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < 9.99999999999999909e-308

   1. Initial program 14.4

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 14.3

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 14.4	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 14.3	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 10.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}} \]

   if 9.99999999999999909e-308 < h

   1. Initial program 14.5

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

   2. Simplified 14.5

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}} \]
      Proof

      [Start] 14.5	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      times-frac [=>] 14.5	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 11.4

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}} \]

   4. Applied egg-rr 8.7

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \sqrt{h}}{\ell} \cdot \left(\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \sqrt{h}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq 10^{-307}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right) \cdot \sqrt{h}}{\ell} \cdot \left(\sqrt{h} \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot -0.5\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.6
Cost	21060

\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{h}{\ell} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	10.6
Cost	8392

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(D \cdot \frac{M}{d}\right) \cdot \left(\frac{D}{d} \cdot \left(M \cdot -0.5\right)\right)}{2 \cdot \frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	14.1
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+43}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-44}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{h \cdot D}{\ell} \cdot \left(D \cdot \frac{-0.25}{\frac{d}{M \cdot \frac{M}{d}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	10.4
Cost	8264

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1 \cdot 10^{-186}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot t_0}{\frac{\ell}{h} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 5
Error	14.2
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))