Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 20.4

Time: 30.2s

Precision: binary64

Cost: 11076

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \leq 10^{+240}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{d \cdot \frac{\frac{\frac{d}{h}}{M}}{D}}\\ \end{array} \]

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
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))) 1e+240)
     (* (/ (/ c0 2.0) w) (* 2.0 (* (/ (/ d D) (* w h)) (* c0 (/ d D)))))
     (* 0.25 (/ (* D M) (* d (/ (/ (/ d h) M) D)))))))

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 t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= 1e+240) {
		tmp = ((c0 / 2.0) / w) * (2.0 * (((d / D) / (w * h)) * (c0 * (d / D))));
	} else {
		tmp = 0.25 * ((D * M) / (d * (((d / h) / M) / D)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    if (((c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))) <= 1d+240) then
        tmp = ((c0 / 2.0d0) / w) * (2.0d0 * (((d_1 / d) / (w * h)) * (c0 * (d_1 / d))))
    else
        tmp = 0.25d0 * ((d * m) / (d_1 * (((d_1 / h) / m) / d)))
    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 t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= 1e+240) {
		tmp = ((c0 / 2.0) / w) * (2.0 * (((d / D) / (w * h)) * (c0 * (d / D))));
	} else {
		tmp = 0.25 * ((D * M) / (d * (((d / h) / M) / D)));
	}
	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):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= 1e+240:
		tmp = ((c0 / 2.0) / w) * (2.0 * (((d / D) / (w * h)) * (c0 * (d / D))))
	else:
		tmp = 0.25 * ((D * M) / (d * (((d / h) / M) / D)))
	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)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= 1e+240)
		tmp = Float64(Float64(Float64(c0 / 2.0) / w) * Float64(2.0 * Float64(Float64(Float64(d / D) / Float64(w * h)) * Float64(c0 * Float64(d / D)))));
	else
		tmp = Float64(0.25 * Float64(Float64(D * M) / Float64(d * Float64(Float64(Float64(d / h) / M) / D))));
	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)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= 1e+240)
		tmp = ((c0 / 2.0) / w) * (2.0 * (((d / D) / (w * h)) * (c0 * (d / D))));
	else
		tmp = 0.25 * ((D * M) / (d * (((d / h) / M) / D)));
	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_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+240], N[(N[(N[(c0 / 2.0), $MachinePrecision] / w), $MachinePrecision] * N[(2.0 * N[(N[(N[(d / D), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(c0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(D * M), $MachinePrecision] / N[(d * N[(N[(N[(d / h), $MachinePrecision] / M), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < 1.00000000000000001e240

   1. Initial program 35.2

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

   2. Simplified 53.4

      \[\leadsto \color{blue}{\frac{\frac{c0}{2}}{w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{w \cdot \left(h \cdot \left(D \cdot D\right)\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot {\left(\frac{d}{D}\right)}^{4}, M \cdot \left(-M\right)\right)}\right)} \]
      Proof
      (*.f64 (/.f64 (/.f64 c0 2) w) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 w (*.f64 h (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 c0 (*.f64 2 w))) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 w (*.f64 h (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) 4)) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (pow.f64 (/.f64 d D) (Rewrite<= metadata-eval (+.f64 3 1)))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (/.f64 d D) 3) (/.f64 d D)))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (/.f64 d D))) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 d D)) (/.f64 d D))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (*.f64 (/.f64 d D) (/.f64 d D))))) (*.f64 M (neg.f64 M)))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M (neg.f64 M)))))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 M M))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 c0 (*.f64 w h))) (*.f64 (/.f64 (*.f64 d d) (*.f64 D D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 8 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (Rewrite<= swap-sqr_binary64 (*.f64 (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D))))) (*.f64 M M))))): 0 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 (/.f64 c0 (*.f64 w h)) (/.f64 (*.f64 d d) (*.f64 D D)))) (*.f64 M M))))): 5 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))))) (*.f64 M M))))): 2 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 d d) (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))): 2 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 c0 (*.f64 (*.f64 w h) (*.f64 D D))) (*.f64 d d))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))): 1 points increase in error, 6 points decrease in error

   3. Taylor expanded in d around inf 40.0

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   4. Simplified 41.0

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \left(c0 \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{w \cdot h}\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 c0 (/.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (*.f64 w h)))): 41 points increase in error, 17 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (/.f64 (/.f64 (pow.f64 d 2) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 c0 (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))): 8 points increase in error, 9 points decrease in error
      (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 c0 (pow.f64 d 2)) (*.f64 (pow.f64 D 2) (*.f64 w h))))): 18 points increase in error, 9 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 d 2) c0)) (*.f64 (pow.f64 D 2) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 32.4

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \color{blue}{\frac{\frac{d}{D} \cdot c0}{\left(w \cdot h\right) \cdot \frac{D}{d}}}\right) \]

   6. Applied egg-rr 32.4

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \color{blue}{\left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right)}\right) \]

   if 1.00000000000000001e240 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 63.9

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

   2. Applied egg-rr 59.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \sqrt{{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}^{2} - M \cdot M}\right)}\right)}^{3}} \]

   3. Taylor expanded in c0 around -inf 63.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot {c0}^{2}}{w} + 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]

   4. Simplified 29.1

      \[\leadsto \color{blue}{0.25 \cdot \left(\frac{D \cdot D}{d} \cdot \frac{M \cdot \left(M \cdot h\right)}{d}\right)} \]
      Proof
      (*.f64 1/4 (*.f64 (/.f64 (*.f64 D D) d) (/.f64 (*.f64 M (*.f64 M h)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) d) (/.f64 (*.f64 M (*.f64 M h)) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h)) d))): 17 points increase in error, 2 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (*.f64 (/.f64 (pow.f64 D 2) d) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2))) d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 h (pow.f64 M 2))) (*.f64 d d)))): 29 points increase in error, 6 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) (*.f64 d d))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= metadata-eval (*.f64 -1/2 0)) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (Rewrite<= div0_binary64 (/.f64 0 (/.f64 w (pow.f64 c0 2))))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 44 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 87 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))) (/.f64 w (pow.f64 c0 2)))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))) (*.f64 -1 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) (pow.f64 c0 2)) w))) (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 0 points increase in error, 1 points decrease in error

   5. Applied egg-rr 19.8

      \[\leadsto 0.25 \cdot \color{blue}{\frac{M \cdot D}{\frac{\frac{d}{M}}{h} \cdot \frac{d}{D}}} \]

   6. Taylor expanded in d around 0 25.0

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

   7. Simplified 18.2

      \[\leadsto 0.25 \cdot \frac{M \cdot D}{\color{blue}{d \cdot \frac{\frac{\frac{d}{h}}{M}}{D}}} \]
      Proof
      (*.f64 d (/.f64 (/.f64 (/.f64 d h) M) D)): 0 points increase in error, 0 points decrease in error
      (*.f64 d (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 h M))) D)): 25 points increase in error, 24 points decrease in error
      (*.f64 d (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 (*.f64 h M) D)))): 22 points increase in error, 18 points decrease in error
      (*.f64 d (/.f64 d (Rewrite<= *-commutative_binary64 (*.f64 D (*.f64 h M))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 d (*.f64 D (*.f64 h M))) d)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 d d) (*.f64 D (*.f64 h M)))): 35 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 D (*.f64 h M))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 d 2) (*.f64 D (Rewrite=> *-commutative_binary64 (*.f64 M h)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 20.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 10^{+240}:\\ \;\;\;\;\frac{\frac{c0}{2}}{w} \cdot \left(2 \cdot \left(\frac{\frac{d}{D}}{w \cdot h} \cdot \left(c0 \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{d \cdot \frac{\frac{\frac{d}{h}}{M}}{D}}\\ \end{array} \]

Alternatives

Alternative 1
Error	22.3
Cost	1608

\[\begin{array}{l} \mathbf{if}\;d \cdot d \leq 4 \cdot 10^{+177}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{d \cdot \frac{\frac{\frac{d}{h}}{M}}{D}}\\ \mathbf{elif}\;d \cdot d \leq 5 \cdot 10^{+213}:\\ \;\;\;\;\frac{d \cdot \left(\frac{d}{D} \cdot \frac{c0}{w \cdot \frac{D}{c0}}\right)}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot M}{\frac{d}{D} \cdot \frac{d}{M}}\right)\\ \end{array} \]

Alternative 2
Error	20.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;c0 \leq -5.6 \cdot 10^{+63}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{elif}\;c0 \leq -3.8 \cdot 10^{-268}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot M}{\frac{d}{D} \cdot \frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{\frac{d}{D \cdot \left(h \cdot \frac{M}{d}\right)}}\\ \end{array} \]

Alternative 3
Error	20.4
Cost	1224

\[\begin{array}{l} \mathbf{if}\;d \leq -3.75 \cdot 10^{-171}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{d \cdot \frac{\frac{\frac{d}{h}}{M}}{D}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+277}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{\frac{d}{D \cdot \left(h \cdot \frac{M}{d}\right)}}\\ \end{array} \]

Alternative 4
Error	19.9
Cost	1224

\[\begin{array}{l} \mathbf{if}\;h \leq -0.022:\\ \;\;\;\;0.25 \cdot \frac{\frac{M \cdot \left(D \cdot M\right)}{\frac{\frac{d}{h}}{D}}}{d}\\ \mathbf{elif}\;h \leq 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{\frac{d}{D \cdot \left(h \cdot \frac{M}{d}\right)}}\\ \end{array} \]

Alternative 5
Error	21.8
Cost	1092

\[\begin{array}{l} t_0 := \frac{d}{h \cdot M}\\ \mathbf{if}\;d \leq 10^{+75}:\\ \;\;\;\;0.25 \cdot \frac{D \cdot M}{\frac{t_0}{\frac{D}{d}}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(D \cdot \frac{\frac{M}{\frac{t_0}{D}}}{d}\right)\\ \end{array} \]

Alternative 6
Error	20.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;w \leq 2.2 \cdot 10^{-194}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot M}{\frac{d}{D} \cdot \frac{d}{M}}\right)\\ \end{array} \]

Alternative 7
Error	20.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;w \leq 6 \cdot 10^{+138}:\\ \;\;\;\;0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{D \cdot M}{d \cdot \frac{d}{M}} \cdot \left(h \cdot D\right)\right)\\ \end{array} \]

Alternative 8
Error	22.4
Cost	960

\[0.25 \cdot \frac{D \cdot M}{\frac{\frac{d}{h \cdot M}}{\frac{D}{d}}} \]

Alternative 9
Error	21.8
Cost	960

\[0.25 \cdot \frac{D \cdot M}{\frac{\frac{d}{M}}{h \cdot \frac{D}{d}}} \]

Alternative 10
Error	20.3
Cost	960

\[0.25 \cdot \left(\frac{M}{\frac{d}{D}} \cdot \frac{D}{\frac{d}{h \cdot M}}\right) \]

Alternative 11
Error	31.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022308 
(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))))))