Henrywood and Agarwal, Equation (13)

Average Error: 59.5 → 19.0

Time: 14.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{0.25}{d} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{\frac{1}{h}}\right)\\ \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \cdot M \leq 10^{+228}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (* (/ 0.25 d) (* (/ (* M D) d) (/ (* M D) (/ 1.0 h))))))
   (if (<= (* M M) 4e-127)
     t_0
     (if (<= (* M M) 1e+228)
       (* (/ 0.25 d) (* D (* (* M M) (* h (/ D d)))))
       t_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 t_0 = (0.25 / d) * (((M * D) / d) * ((M * D) / (1.0 / h)));
	double tmp;
	if ((M * M) <= 4e-127) {
		tmp = t_0;
	} else if ((M * M) <= 1e+228) {
		tmp = (0.25 / d) * (D * ((M * M) * (h * (D / d))));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (0.25d0 / d_1) * (((m * d) / d_1) * ((m * d) / (1.0d0 / h)))
    if ((m * m) <= 4d-127) then
        tmp = t_0
    else if ((m * m) <= 1d+228) then
        tmp = (0.25d0 / d_1) * (d * ((m * m) * (h * (d / d_1))))
    else
        tmp = t_0
    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 = (0.25 / d) * (((M * D) / d) * ((M * D) / (1.0 / h)));
	double tmp;
	if ((M * M) <= 4e-127) {
		tmp = t_0;
	} else if ((M * M) <= 1e+228) {
		tmp = (0.25 / d) * (D * ((M * M) * (h * (D / d))));
	} else {
		tmp = t_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):
	t_0 = (0.25 / d) * (((M * D) / d) * ((M * D) / (1.0 / h)))
	tmp = 0
	if (M * M) <= 4e-127:
		tmp = t_0
	elif (M * M) <= 1e+228:
		tmp = (0.25 / d) * (D * ((M * M) * (h * (D / d))))
	else:
		tmp = t_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)
	t_0 = Float64(Float64(0.25 / d) * Float64(Float64(Float64(M * D) / d) * Float64(Float64(M * D) / Float64(1.0 / h))))
	tmp = 0.0
	if (Float64(M * M) <= 4e-127)
		tmp = t_0;
	elseif (Float64(M * M) <= 1e+228)
		tmp = Float64(Float64(0.25 / d) * Float64(D * Float64(Float64(M * M) * Float64(h * Float64(D / d)))));
	else
		tmp = t_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)
	t_0 = (0.25 / d) * (((M * D) / d) * ((M * D) / (1.0 / h)));
	tmp = 0.0;
	if ((M * M) <= 4e-127)
		tmp = t_0;
	elseif ((M * M) <= 1e+228)
		tmp = (0.25 / d) * (D * ((M * M) * (h * (D / d))));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(N[(0.25 / d), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(M * M), $MachinePrecision], 4e-127], t$95$0, If[LessEqual[N[(M * M), $MachinePrecision], 1e+228], N[(N[(0.25 / d), $MachinePrecision] * N[(D * N[(N[(M * M), $MachinePrecision] * N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

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 (*.f64 M M) < 4.0000000000000001e-127 or 9.9999999999999992e227 < (*.f64 M M)

   1. Initial program 58.5

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

   2. Taylor expanded in c0 around -inf 42.2

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

   3. Simplified 39.4

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

   4. Taylor expanded in c0 around 0 36.5

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

   5. Simplified 33.8

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

   6. Applied egg-rr 22.5

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

   7. Applied egg-rr 19.0

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

   if 4.0000000000000001e-127 < (*.f64 M M) < 9.9999999999999992e227

   1. Initial program 62.2

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

   2. Taylor expanded in c0 around -inf 41.3

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

   3. Simplified 38.5

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

   4. Taylor expanded in c0 around 0 32.4

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

   5. Simplified 26.8

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

   6. Applied egg-rr 25.2

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

   7. Taylor expanded in D around 0 29.7

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

   8. Simplified 19.0

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

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 4 \cdot 10^{-127}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{\frac{1}{h}}\right)\\ \mathbf{elif}\;M \cdot M \leq 10^{+228}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(D \cdot \left(\left(M \cdot M\right) \cdot \left(h \cdot \frac{D}{d}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{d} \cdot \left(\frac{M \cdot D}{d} \cdot \frac{M \cdot D}{\frac{1}{h}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))))