Henrywood and Agarwal, Equation (9a)

Average Error: 21.99% → 13.75%

Time: 17.9s

Precision: binary64

Cost: 15236

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot 0.5}{\frac{d}{D}} \cdot \frac{\frac{M \cdot -0.5}{\frac{d}{D}}}{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell \cdot \left(d \cdot \frac{2}{D}\right)}{M \cdot h}}}\\ \end{array} \]

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
 (if (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) 1e+293)
   (*
    w0
    (sqrt
     (+ 1.0 (* (/ (* M 0.5) (/ d D)) (/ (/ (* M -0.5) (/ d D)) (/ l h))))))
   (*
    w0
    (sqrt
     (+ 1.0 (/ (* M (* (/ D d) -0.5)) (/ (* l (* d (/ 2.0 D))) (* M h))))))))

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 tmp;
	if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 1e+293) {
		tmp = w0 * sqrt((1.0 + (((M * 0.5) / (d / D)) * (((M * -0.5) / (d / D)) / (l / h)))));
	} else {
		tmp = w0 * sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l * (d * (2.0 / D))) / (M * h)))));
	}
	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) :: tmp
    if ((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))) <= 1d+293) then
        tmp = w0 * sqrt((1.0d0 + (((m * 0.5d0) / (d_1 / d)) * (((m * (-0.5d0)) / (d_1 / d)) / (l / h)))))
    else
        tmp = w0 * sqrt((1.0d0 + ((m * ((d / d_1) * (-0.5d0))) / ((l * (d_1 * (2.0d0 / d))) / (m * h)))))
    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 tmp;
	if ((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 1e+293) {
		tmp = w0 * Math.sqrt((1.0 + (((M * 0.5) / (d / D)) * (((M * -0.5) / (d / D)) / (l / h)))));
	} else {
		tmp = w0 * Math.sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l * (d * (2.0 / D))) / (M * h)))));
	}
	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):
	tmp = 0
	if (1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 1e+293:
		tmp = w0 * math.sqrt((1.0 + (((M * 0.5) / (d / D)) * (((M * -0.5) / (d / D)) / (l / h)))))
	else:
		tmp = w0 * math.sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l * (d * (2.0 / D))) / (M * h)))))
	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)
	tmp = 0.0
	if (Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))) <= 1e+293)
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(M * 0.5) / Float64(d / D)) * Float64(Float64(Float64(M * -0.5) / Float64(d / D)) / Float64(l / h))))));
	else
		tmp = Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(M * Float64(Float64(D / d) * -0.5)) / Float64(Float64(l * Float64(d * Float64(2.0 / D))) / Float64(M * h))))));
	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)
	tmp = 0.0;
	if ((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))) <= 1e+293)
		tmp = w0 * sqrt((1.0 + (((M * 0.5) / (d / D)) * (((M * -0.5) / (d / D)) / (l / h)))));
	else
		tmp = w0 * sqrt((1.0 + ((M * ((D / d) * -0.5)) / ((l * (d * (2.0 / D))) / (M * h)))));
	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_] := If[LessEqual[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], 1e+293], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * -0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision] / N[(l / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(w0 * N[Sqrt[N[(1.0 + N[(N[(M * N[(N[(D / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(d * N[(2.0 / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 9.9999999999999992e292

   1. Initial program 0.45

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

   2. Applied egg-rr 5.46

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

   3. Applied egg-rr 0.91

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M \cdot 0.5}{\frac{d}{D}}}{\frac{\ell}{h}} \cdot \frac{M \cdot 0.5}{\frac{d}{D}}}} \]

   if 9.9999999999999992e292 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

   1. Initial program 98.98

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

   2. Applied egg-rr 84.19

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

   3. Applied egg-rr 59.66

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

4. Final simplification 13.75

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 10^{+293}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot 0.5}{\frac{d}{D}} \cdot \frac{\frac{M \cdot -0.5}{\frac{d}{D}}}{\frac{\ell}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell \cdot \left(d \cdot \frac{2}{D}\right)}{M \cdot h}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.19%
Cost	8009

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\ell \leq -8 \cdot 10^{-101} \lor \neg \left(\ell \leq 7.5 \cdot 10^{-145}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{\frac{\ell}{h} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	20.91%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{+165}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \left(\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right) \cdot -0.25}\\ \mathbf{elif}\;h \leq 5 \cdot 10^{-120}:\\ \;\;\;\;w0 \cdot \sqrt{1 + D \cdot \left(M \cdot \frac{\frac{\frac{h}{\ell}}{d} \cdot \left(M \cdot -0.25\right)}{\frac{d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	17.7%
Cost	8008

\[\begin{array}{l} t_0 := D \cdot \frac{M}{d}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-93}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot \frac{h}{\ell \cdot 4}}{\frac{\frac{d}{D}}{M}}}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-151}:\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0}{\frac{\ell}{h} \cdot 4}}\\ \end{array} \]

Alternative 4
Error	14.66%
Cost	7872

\[w0 \cdot \sqrt{1 + \left(h \cdot \frac{D \cdot \left(0.5 \cdot \frac{M}{d}\right)}{\ell}\right) \cdot \frac{M \cdot -0.5}{\frac{d}{D}}} \]

Alternative 5
Error	21.56%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023102 
(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))))))