Henrywood and Agarwal, Equation (9a)

Average Error: 14.3 → 10.3

Time: 20.4s

Precision: binary64

Cost: 14084

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-71}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{4}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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 (<= (/ h l) -1e-71)
   (* w0 (sqrt (- 1.0 (/ h (* (/ 4.0 (pow (* M (/ D d)) 2.0)) l)))))
   w0))

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 ((h / l) <= -1e-71) {
		tmp = w0 * sqrt((1.0 - (h / ((4.0 / pow((M * (D / d)), 2.0)) * l))));
	} else {
		tmp = w0;
	}
	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 ((h / l) <= (-1d-71)) then
        tmp = w0 * sqrt((1.0d0 - (h / ((4.0d0 / ((m * (d / d_1)) ** 2.0d0)) * l))))
    else
        tmp = w0
    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 ((h / l) <= -1e-71) {
		tmp = w0 * Math.sqrt((1.0 - (h / ((4.0 / Math.pow((M * (D / d)), 2.0)) * l))));
	} else {
		tmp = w0;
	}
	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 (h / l) <= -1e-71:
		tmp = w0 * math.sqrt((1.0 - (h / ((4.0 / math.pow((M * (D / d)), 2.0)) * l))))
	else:
		tmp = w0
	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(h / l) <= -1e-71)
		tmp = Float64(w0 * sqrt(Float64(1.0 - Float64(h / Float64(Float64(4.0 / (Float64(M * Float64(D / d)) ^ 2.0)) * l)))));
	else
		tmp = w0;
	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 ((h / l) <= -1e-71)
		tmp = w0 * sqrt((1.0 - (h / ((4.0 / ((M * (D / d)) ^ 2.0)) * l))));
	else
		tmp = w0;
	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[(h / l), $MachinePrecision], -1e-71], N[(w0 * N[Sqrt[N[(1.0 - N[(h / N[(N[(4.0 / N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], w0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 h l) < -9.9999999999999992e-72

   1. Initial program 22.6

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

   2. Simplified 22.8

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

      [Start] 22.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational.json-simplify-2 [=>] 22.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational.json-simplify-49 [=>] 22.8	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Applied egg-rr 17.4

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

   4. Applied egg-rr 17.4

      \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\color{blue}{\frac{4}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \ell}}} \]

   if -9.9999999999999992e-72 < (/.f64 h l)

   1. Initial program 10.0

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

   2. Simplified 10.1

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

      [Start] 10.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational.json-simplify-2 [=>] 10.0	\[ w0 \cdot \sqrt{1 - {\left(\frac{\color{blue}{D \cdot M}}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational.json-simplify-49 [=>] 10.1	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 6.6

      \[\leadsto \color{blue}{w0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-71}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\frac{4}{{\left(M \cdot \frac{D}{d}\right)}^{2}} \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	21188

\[\begin{array}{l} t_0 := 1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{+264}:\\ \;\;\;\;w0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	12.6
Cost	14352

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 - {\left(\frac{M}{d} \cdot \frac{D}{2}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;h \leq -3.8 \cdot 10^{+101}:\\ \;\;\;\;w0\\ \mathbf{elif}\;h \leq 4.4 \cdot 10^{-297}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;h \leq 10^{-160}:\\ \;\;\;\;w0\\ \mathbf{elif}\;h \leq 1.9 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	11.8
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	13.7
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023075 
(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))))))