Henrywood and Agarwal, Equation (9a)

Average Error: 23.1% → 13.98%

Time: 17.0s

Precision: binary64

Cost: 7744

Math

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

↓

\[w0 \cdot \sqrt{1 - \frac{h}{\frac{d}{M \cdot D} \cdot \ell} \cdot \frac{\left(M \cdot D\right) \cdot 0.25}{d}} \]

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
 (* w0 (sqrt (- 1.0 (* (/ h (* (/ d (* M D)) l)) (/ (* (* M D) 0.25) d))))))

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) {
	return w0 * sqrt((1.0 - ((h / ((d / (M * D)) * l)) * (((M * D) * 0.25) / d))));
}

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
    code = w0 * sqrt((1.0d0 - ((h / ((d_1 / (m * d)) * l)) * (((m * d) * 0.25d0) / d_1))))
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) {
	return w0 * Math.sqrt((1.0 - ((h / ((d / (M * D)) * l)) * (((M * D) * 0.25) / d))));
}

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):
	return w0 * math.sqrt((1.0 - ((h / ((d / (M * D)) * l)) * (((M * D) * 0.25) / d))))

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)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(h / Float64(Float64(d / Float64(M * D)) * l)) * Float64(Float64(Float64(M * D) * 0.25) / d)))))
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 = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((h / ((d / (M * D)) * l)) * (((M * D) * 0.25) / d))));
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_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[(h / N[(N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] * 0.25), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.1

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

2. Applied egg-rr 37.89

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

3. Applied egg-rr 20.61

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{d} \cdot \frac{h}{\ell}\right) \cdot \frac{0.25 \cdot \left(M \cdot D\right)}{d}}} \]

4. Applied egg-rr 13.98

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{h}{\frac{d}{M \cdot D} \cdot \ell}} \cdot \frac{0.25 \cdot \left(M \cdot D\right)}{d}} \]

5. Final simplification 13.98

   \[\leadsto w0 \cdot \sqrt{1 - \frac{h}{\frac{d}{M \cdot D} \cdot \ell} \cdot \frac{\left(M \cdot D\right) \cdot 0.25}{d}} \]

Alternatives

Alternative 1
Error	15.96%
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{+278}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \leq 4 \cdot 10^{+287}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(M \cdot D\right) \cdot 0.25}{d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot D}{\frac{d}{M \cdot \frac{h}{d}} \cdot \frac{\ell}{D}} \cdot -0.25}\\ \end{array} \]

Alternative 2
Error	21.49%
Cost	8141

\[\begin{array}{l} \mathbf{if}\;D \leq -2.4 \cdot 10^{-171} \lor \neg \left(D \leq 7.5 \cdot 10^{-148}\right) \land D \leq 3.2 \cdot 10^{+92}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot D}{\frac{d}{M \cdot \frac{h}{d}} \cdot \frac{\ell}{D}} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	23.03%
Cost	8008

\[\begin{array}{l} \mathbf{if}\;M \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;w0\\ \mathbf{elif}\;M \leq -7.4 \cdot 10^{-93}:\\ \;\;\;\;w0 \cdot \sqrt{1 + -0.25 \cdot \left(\frac{D}{\frac{\ell}{D}} \cdot \left(h \cdot \left(\frac{M}{d} \cdot \frac{M}{d}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	22.54%
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023089 
(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))))))