Henrywood and Agarwal, Equation (13)

Average Error: 59.9 → 20.2

Time: 19.5s

Precision: binary64

Math

\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;h \leq -3.7426609267893 \cdot 10^{-310}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {\left(\frac{D \cdot \left(M \cdot \sqrt{h}\right)}{d}\right)}^{2}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= h -3.7426609267893e-310)
   (* 0.25 (/ (* h (/ (pow (* D M) 2.0) d)) d))
   (* 0.25 (pow (/ (* D (* M (sqrt h))) d) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -3.7426609267893e-310) {
		tmp = 0.25 * ((h * (pow((D * M), 2.0) / d)) / d);
	} else {
		tmp = 0.25 * pow(((D * (M * sqrt(h))) / d), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * (((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) + sqrt(((((c0 * (d_1 * d_1)) / ((w * h) * (d * d))) * ((c0 * (d_1 * d_1)) / ((w * h) * (d * d)))) - (m * m))))
end function

↓

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (h <= (-3.7426609267893d-310)) then
        tmp = 0.25d0 * ((h * (((d * m) ** 2.0d0) / d_1)) / d_1)
    else
        tmp = 0.25d0 * (((d * (m * sqrt(h))) / d_1) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + Math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (h <= -3.7426609267893e-310) {
		tmp = 0.25 * ((h * (Math.pow((D * M), 2.0) / d)) / d);
	} else {
		tmp = 0.25 * Math.pow(((D * (M * Math.sqrt(h))) / d), 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + math.sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))))

↓

def code(c0, w, h, D, d, M):
	tmp = 0
	if h <= -3.7426609267893e-310:
		tmp = 0.25 * ((h * (math.pow((D * M), 2.0) / d)) / d)
	else:
		tmp = 0.25 * math.pow(((D * (M * math.sqrt(h))) / d), 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) + sqrt(Float64(Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))) - Float64(M * M)))))
end

↓

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (h <= -3.7426609267893e-310)
		tmp = Float64(0.25 * Float64(Float64(h * Float64((Float64(D * M) ^ 2.0) / d)) / d));
	else
		tmp = Float64(0.25 * (Float64(Float64(D * Float64(M * sqrt(h))) / d) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
end

↓

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (h <= -3.7426609267893e-310)
		tmp = 0.25 * ((h * (((D * M) ^ 2.0) / d)) / d);
	else
		tmp = 0.25 * (((D * (M * sqrt(h))) / d) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[h, -3.7426609267893e-310], N[(0.25 * N[(N[(h * N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Power[N[(N[(D * N[(M * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

1. Split input into 2 regimes

2. if h < -3.742660926789291e-310

   1. Initial program 60.0

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 35.9

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   3. Applied egg-rr 24.0

      \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{1}{d} \cdot \frac{h \cdot {\left(D \cdot M\right)}^{2}}{d}\right)} \]

   4. Applied egg-rr 22.8

      \[\leadsto 0.25 \cdot \left(\frac{1}{d} \cdot \color{blue}{\left(h \cdot \frac{1}{\frac{d}{{\left(D \cdot M\right)}^{2}}}\right)}\right) \]

   5. Applied egg-rr 22.7

      \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{h \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}}{d}\right)}^{1}} \]

   if -3.742660926789291e-310 < h

   1. Initial program 59.8

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 35.6

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   3. Applied egg-rr 17.7

      \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{D \cdot \left(M \cdot \sqrt{h}\right)}{d}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.7426609267893 \cdot 10^{-310}:\\ \;\;\;\;0.25 \cdot \frac{h \cdot \frac{{\left(D \cdot M\right)}^{2}}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot {\left(\frac{D \cdot \left(M \cdot \sqrt{h}\right)}{d}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022145 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))