Henrywood and Agarwal, Equation (13)

Average Error: 59.6 → 19.9

Time: 26.8s

Precision: binary64

Cost: 11076

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right) \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \frac{0.5}{w}\right)}{\frac{h}{\frac{\frac{d}{w}}{D} \cdot \frac{c0}{\frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{h \cdot D}{d}\right)\right)\\ \end{array} \]

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-170)
     (/ (* c0 (* 2.0 (/ 0.5 w))) (/ h (* (/ (/ d w) D) (/ c0 (/ D d)))))
     (* 0.25 (* (* M (/ D d)) (* M (/ (* h D) 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-170) {
		tmp = (c0 * (2.0 * (0.5 / w))) / (h / (((d / w) / D) * (c0 / (D / d))));
	} else {
		tmp = 0.25 * ((M * (D / d)) * (M * ((h * D) / 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-170)) then
        tmp = (c0 * (2.0d0 * (0.5d0 / w))) / (h / (((d_1 / w) / d) * (c0 / (d / d_1))))
    else
        tmp = 0.25d0 * ((m * (d / d_1)) * (m * ((h * d) / d_1)))
    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-170) {
		tmp = (c0 * (2.0 * (0.5 / w))) / (h / (((d / w) / D) * (c0 / (D / d))));
	} else {
		tmp = 0.25 * ((M * (D / d)) * (M * ((h * D) / 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-170:
		tmp = (c0 * (2.0 * (0.5 / w))) / (h / (((d / w) / D) * (c0 / (D / d))))
	else:
		tmp = 0.25 * ((M * (D / d)) * (M * ((h * D) / 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-170)
		tmp = Float64(Float64(c0 * Float64(2.0 * Float64(0.5 / w))) / Float64(h / Float64(Float64(Float64(d / w) / D) * Float64(c0 / Float64(D / d)))));
	else
		tmp = Float64(0.25 * Float64(Float64(M * Float64(D / d)) * Float64(M * Float64(Float64(h * D) / 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-170)
		tmp = (c0 * (2.0 * (0.5 / w))) / (h / (((d / w) / D) * (c0 / (D / d))));
	else
		tmp = 0.25 * ((M * (D / d)) * (M * ((h * D) / 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-170], N[(N[(c0 * N[(2.0 * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(h / N[(N[(N[(d / w), $MachinePrecision] / D), $MachinePrecision] * N[(c0 / N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[(M * N[(N[(h * D), $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))))) < -9.99999999999999983e-171

   1. Initial program 48.8

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

   2. Taylor expanded in c0 around inf 43.4

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

   3. Simplified 40.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{\frac{d}{w} \cdot \frac{d \cdot c0}{D \cdot D}}{h}\right)} \]
      Proof
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (*.f64 (/.f64 d w) (/.f64 (*.f64 d c0) (*.f64 D D))) h))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (*.f64 (/.f64 d w) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 c0 d)) (*.f64 D D))) h))): 12 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 d w) h) (/.f64 (*.f64 c0 d) (*.f64 D D)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 w h))) (/.f64 (*.f64 c0 d) (*.f64 D D))))): 0 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 d (/.f64 (*.f64 c0 d) (*.f64 D D))) (*.f64 w h))))): 12 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 c0 d) (*.f64 D D)) d)) (*.f64 w h)))): 11 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 c0 d) d) (*.f64 D D))) (*.f64 w h)))): 0 points increase in error, 12 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 c0 (*.f64 d d))) (*.f64 D D)) (*.f64 w h)))): 12 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (/.f64 (*.f64 c0 (Rewrite<= unpow2_binary64 (pow.f64 d 2))) (*.f64 D D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 d 2) c0)) (*.f64 D D)) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (/.f64 (/.f64 (*.f64 (pow.f64 d 2) c0) (Rewrite<= unpow2_binary64 (pow.f64 D 2))) (*.f64 w h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (*.f64 2 (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 (pow.f64 d 2) c0) (*.f64 (pow.f64 D 2) (*.f64 w h)))))): 0 points increase in error, 6 points decrease in error

   4. Applied egg-rr 33.7

      \[\leadsto \color{blue}{\frac{c0 \cdot \left(\frac{0.5}{w} \cdot 2\right)}{\frac{h}{\frac{\frac{d}{w}}{D} \cdot \frac{c0}{\frac{D}{d}}}}} \]

   if -9.99999999999999983e-171 < (*.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 60.3

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

   2. Simplified 62.4

      \[\leadsto \color{blue}{\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(d \cdot d, \frac{c0}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}, \sqrt{\mathsf{fma}\left(\frac{c0}{w \cdot h}, \frac{c0}{w \cdot h} \cdot \left(\frac{d}{D} \cdot {\left(\frac{d}{D}\right)}^{3}\right), -M \cdot M\right)}\right)} \]
      Proof
      (*.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 d D) (pow.f64 (/.f64 d D) 3))) (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 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 d D) (Rewrite<= cube-unmult_binary64 (*.f64 (/.f64 d D) (*.f64 (/.f64 d D) (/.f64 d D)))))) (neg.f64 (*.f64 M M)))))): 6 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 d D) (*.f64 (/.f64 d D) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D)))))) (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 (fma.f64 (/.f64 c0 (*.f64 w h)) (*.f64 (/.f64 c0 (*.f64 w h)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 d D) (/.f64 d D)) (/.f64 (*.f64 d d) (*.f64 D D))))) (neg.f64 (*.f64 M M)))))): 0 points increase in error, 6 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<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))) (/.f64 (*.f64 d d) (*.f64 D D)))) (neg.f64 (*.f64 M M)))))): 6 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))))): 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 (-.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))))): 4 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 (*.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))))): 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 (*.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))))): 0 points increase in error, 0 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)))))): 0 points increase in error, 0 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))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in c0 around -inf 60.1

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

   4. Simplified 33.2

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0, c0, \frac{0.5}{c0} \cdot \frac{w \cdot \left(M \cdot \left(M \cdot h\right)\right)}{{\left(\frac{d}{D}\right)}^{2}}\right)} \]
      Proof
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (*.f64 M (*.f64 M h))) (pow.f64 (/.f64 d D) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 M M) h))) (pow.f64 (/.f64 d D) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) (pow.f64 (/.f64 d D) 2))))): 13 points increase in error, 9 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (Rewrite<= *-commutative_binary64 (*.f64 h (pow.f64 M 2)))) (pow.f64 (/.f64 d D) 2))))): 24 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (Rewrite=> unpow2_binary64 (*.f64 (/.f64 d D) (/.f64 d D))))))): 9 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 D D))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (/.f64 (*.f64 d d) (Rewrite<= unpow2_binary64 (pow.f64 D 2))))))): 15 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) (pow.f64 D 2)) (*.f64 d d)))))): 0 points increase in error, 9 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (*.f64 (/.f64 1/2 c0) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2))))) (*.f64 d d))))): 0 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2))))) (*.f64 c0 (*.f64 d d)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (/.f64 (*.f64 1/2 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2))))) (*.f64 c0 (Rewrite<= unpow2_binary64 (pow.f64 d 2)))))): 15 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (/.f64 (*.f64 1/2 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2))))) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 d 2) c0))))): 0 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (fma.f64 0 c0 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 0 c0) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0)))))): 15 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 0)) c0) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0))))): 0 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 (neg.f64 (Rewrite<= mul0-lft_binary64 (*.f64 0 (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h)))))) c0) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0))))): 0 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 (neg.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 D 2) (*.f64 w h))))) c0) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0))))): 0 points increase in error, 9 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 (neg.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))))))) c0) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.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))))) c0))) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0))))): 9 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.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))))) c0))) (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 h (pow.f64 M 2)))) (*.f64 (pow.f64 d 2) c0))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 -1 (*.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))))) c0)) (*.f64 1/2 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (/.f64 (*.f64 w (*.f64 h (pow.f64 M 2))) c0)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 -1 (*.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))))) c0)) (*.f64 1/2 (*.f64 (/.f64 (pow.f64 D 2) (pow.f64 d 2)) (/.f64 (*.f64 w (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h))) c0))))): 15 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (*.f64 -1 (*.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))))) c0)) (*.f64 1/2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0)))))): 0 points increase in error, 15 points decrease in error
      (*.f64 (/.f64 c0 (*.f64 2 w)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 w (*.f64 (pow.f64 M 2) h))) (*.f64 (pow.f64 d 2) c0))) (*.f64 -1 (*.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))))) c0))))): 11 points increase in error, 0 points decrease in error

   5. Taylor expanded in c0 around 0 34.3

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

   6. Simplified 21.9

      \[\leadsto \color{blue}{0.25 \cdot \frac{D}{\frac{\frac{d}{M} \cdot \frac{d}{M \cdot h}}{D}}} \]
      Proof
      (*.f64 1/4 (/.f64 D (/.f64 (*.f64 (/.f64 d M) (/.f64 d (*.f64 M h))) D))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 M (*.f64 M h)))) D))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 M (*.f64 M h))) D))): 8 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (/.f64 (pow.f64 d 2) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 M M) h))) D))): 8 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (/.f64 (pow.f64 d 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 M 2)) h)) D))): 0 points increase in error, 8 points decrease in error
      (*.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 D D) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 M 2) h))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (/.f64 (pow.f64 d 2) (*.f64 (pow.f64 M 2) h)))): 8 points increase in error, 0 points decrease in error
      (*.f64 1/4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (pow.f64 d 2)))): 1 points increase in error, 7 points decrease in error

   7. Taylor expanded in d around 0 30.6

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

   8. Simplified 21.5

      \[\leadsto 0.25 \cdot \frac{D}{\color{blue}{\frac{{\left(\frac{d}{M}\right)}^{2}}{D \cdot h}}} \]
      Proof
      (*.f64 1/4 (/.f64 D (/.f64 (pow.f64 (/.f64 d M) 2) (*.f64 D h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 d M) (/.f64 d M))) (*.f64 D h)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (/.f64 d M) D) (/.f64 (/.f64 d M) h))))): 13 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (*.f64 (/.f64 (/.f64 d M) D) (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 M h)))))): 16 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (*.f64 (/.f64 (/.f64 d M) D) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 d h) M))))): 0 points increase in error, 16 points decrease in error
      (*.f64 1/4 (/.f64 D (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (/.f64 d M) (/.f64 (/.f64 d h) M)) D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (/.f64 d M) (/.f64 d h)) M)) D))): 11 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 d M) M) (/.f64 d h))) D))): 7 points increase in error, 9 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 d (*.f64 M M))) (/.f64 d h)) D))): 0 points increase in error, 16 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 d d) (*.f64 (*.f64 M M) h))) D))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (/.f64 (*.f64 d d) (Rewrite=> *-commutative_binary64 (*.f64 h (*.f64 M M)))) D))): 12 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (/.f64 (*.f64 d d) (*.f64 h (Rewrite<= unpow2_binary64 (pow.f64 M 2)))) D))): 0 points increase in error, 12 points decrease in error
      (*.f64 1/4 (/.f64 D (Rewrite<= associate-/r*_binary64 (/.f64 (*.f64 d d) (*.f64 (*.f64 h (pow.f64 M 2)) D))))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 d 2)) (*.f64 (*.f64 h (pow.f64 M 2)) D)))): 0 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (pow.f64 d 2) (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 M 2) h)) D)))): 16 points increase in error, 0 points decrease in error
      (*.f64 1/4 (/.f64 D (/.f64 (pow.f64 d 2) (Rewrite<= *-commutative_binary64 (*.f64 D (*.f64 (pow.f64 M 2) h)))))): 0 points increase in error, 16 points decrease in error

   9. Applied egg-rr 19.1

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

4. Final simplification 19.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \frac{0.5}{w}\right)}{\frac{h}{\frac{\frac{d}{w}}{D} \cdot \frac{c0}{\frac{D}{d}}}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{h \cdot D}{d}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;M \cdot M \leq 5 \cdot 10^{+289}:\\ \;\;\;\;0.25 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \left(M \cdot \frac{h \cdot D}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{D}{\frac{d}{D \cdot M} \cdot \frac{\frac{d}{M}}{h}}\\ \end{array} \]

Alternative 2
Error	21.0
Cost	960

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

Alternative 3
Error	31.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022349 
(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))))))