Henrywood and Agarwal, Equation (9a)

Average Error: 13.7 → 9.3

Time: 13.1s

Precision: binary64

Cost: 27588

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_0 \leq 10^{+149}:\\ \;\;\;\;w0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
   (if (<= t_0 1e+149) (* w0 t_0) 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 t_0 = sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 1e+149) {
		tmp = w0 * t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
    if (t_0 <= 1d+149) then
        tmp = w0 * t_0
    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 t_0 = Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
	double tmp;
	if (t_0 <= 1e+149) {
		tmp = w0 * t_0;
	} 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):
	t_0 = math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
	tmp = 0
	if t_0 <= 1e+149:
		tmp = w0 * t_0
	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)
	t_0 = sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 1e+149)
		tmp = Float64(w0 * t_0);
	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)
	t_0 = sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
	tmp = 0.0;
	if (t_0 <= 1e+149)
		tmp = w0 * t_0;
	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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 1e+149], N[(w0 * t$95$0), $MachinePrecision], w0]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))) < 1.00000000000000005e149

   1. Initial program 0.1

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

   if 1.00000000000000005e149 < (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))

   1. Initial program 63.6

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

   2. Simplified 63.6

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

      [Start] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{d \cdot 2}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      metadata-eval [<=] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d \cdot \color{blue}{\left(1 + 1\right)}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{1 \cdot d + d \cdot 1}}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{d \cdot 1} + d \cdot 1}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{\color{blue}{d} + d \cdot 1}\right)}^{2} \cdot \frac{h}{\ell}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 63.6	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{d + \color{blue}{d}}\right)}^{2} \cdot \frac{h}{\ell}} \]

   3. Taylor expanded in M around 0 43.2

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 10^{+149}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternatives

Alternative 1
Error	13.6
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023090 
(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))))))