Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.2% → 47.1%

Time: 19.3s

Alternatives: 11

Speedup: 156.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

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));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

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]}, 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]]

Initial Program: 24.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

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
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

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));
	return (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

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]}, 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]]

Alternative 1: 47.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(D \cdot w\right) \cdot h\\ \mathbf{if}\;c0 \leq -8 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(\frac{c0}{\left(t\_0 \cdot D\right) \cdot w} \cdot d\right) \cdot d\right) \cdot c0\\ \mathbf{elif}\;c0 \leq -1.4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\left(--0.5\right) \cdot \left(\frac{\left(\left(h \cdot w\right) \cdot D\right) \cdot M}{d} \cdot \frac{M \cdot D}{\left(c0 \cdot c0\right) \cdot d}\right)\right) \cdot c0\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot c0}{\left(\frac{D}{d \cdot c0} \cdot t\_0\right) \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* D w) h)))
   (if (<= c0 -8e+29)
     (* (* (* (/ c0 (* (* t_0 D) w)) d) d) c0)
     (if (<= c0 -1.4e-53)
       (*
        (*
         (* (- -0.5) (* (/ (* (* (* h w) D) M) d) (/ (* M D) (* (* c0 c0) d))))
         c0)
        (/ c0 (* w 2.0)))
       (/ (* d c0) (* (* (/ D (* d c0)) t_0) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * w) * h;
	double tmp;
	if (c0 <= -8e+29) {
		tmp = (((c0 / ((t_0 * D) * w)) * d) * d) * c0;
	} else if (c0 <= -1.4e-53) {
		tmp = ((-(-0.5) * (((((h * w) * D) * M) / d) * ((M * D) / ((c0 * c0) * d)))) * c0) * (c0 / (w * 2.0));
	} else {
		tmp = (d * c0) / (((D / (d * c0)) * t_0) * w);
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (d * w) * h
    if (c0 <= (-8d+29)) then
        tmp = (((c0 / ((t_0 * d) * w)) * d_1) * d_1) * c0
    else if (c0 <= (-1.4d-53)) then
        tmp = ((-(-0.5d0) * (((((h * w) * d) * m) / d_1) * ((m * d) / ((c0 * c0) * d_1)))) * c0) * (c0 / (w * 2.0d0))
    else
        tmp = (d_1 * c0) / (((d / (d_1 * c0)) * t_0) * w)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * w) * h;
	double tmp;
	if (c0 <= -8e+29) {
		tmp = (((c0 / ((t_0 * D) * w)) * d) * d) * c0;
	} else if (c0 <= -1.4e-53) {
		tmp = ((-(-0.5) * (((((h * w) * D) * M) / d) * ((M * D) / ((c0 * c0) * d)))) * c0) * (c0 / (w * 2.0));
	} else {
		tmp = (d * c0) / (((D / (d * c0)) * t_0) * w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (D * w) * h
	tmp = 0
	if c0 <= -8e+29:
		tmp = (((c0 / ((t_0 * D) * w)) * d) * d) * c0
	elif c0 <= -1.4e-53:
		tmp = ((-(-0.5) * (((((h * w) * D) * M) / d) * ((M * D) / ((c0 * c0) * d)))) * c0) * (c0 / (w * 2.0))
	else:
		tmp = (d * c0) / (((D / (d * c0)) * t_0) * w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * w) * h)
	tmp = 0.0
	if (c0 <= -8e+29)
		tmp = Float64(Float64(Float64(Float64(c0 / Float64(Float64(t_0 * D) * w)) * d) * d) * c0);
	elseif (c0 <= -1.4e-53)
		tmp = Float64(Float64(Float64(Float64(-(-0.5)) * Float64(Float64(Float64(Float64(Float64(h * w) * D) * M) / d) * Float64(Float64(M * D) / Float64(Float64(c0 * c0) * d)))) * c0) * Float64(c0 / Float64(w * 2.0)));
	else
		tmp = Float64(Float64(d * c0) / Float64(Float64(Float64(D / Float64(d * c0)) * t_0) * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * w) * h;
	tmp = 0.0;
	if (c0 <= -8e+29)
		tmp = (((c0 / ((t_0 * D) * w)) * d) * d) * c0;
	elseif (c0 <= -1.4e-53)
		tmp = ((-(-0.5) * (((((h * w) * D) * M) / d) * ((M * D) / ((c0 * c0) * d)))) * c0) * (c0 / (w * 2.0));
	else
		tmp = (d * c0) / (((D / (d * c0)) * t_0) * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * w), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[c0, -8e+29], N[(N[(N[(N[(c0 / N[(N[(t$95$0 * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] * d), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[c0, -1.4e-53], N[(N[(N[((--0.5) * N[(N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / N[(N[(c0 * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(D / N[(d * c0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -7.99999999999999931e29

   1. Initial program 30.0%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 36.0

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

   5. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \left(c0 \cdot c0\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.6%

      \[\leadsto \left(d \cdot c0\right) \cdot \color{blue}{\frac{d \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot \left(D \cdot h\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.0%

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

   if -7.99999999999999931e29 < c0 < -1.39999999999999993e-53

   1. Initial program 14.1%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-1 \cdot \left(c0 \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \left(\frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \left(\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      6. mul0-lft N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-1 \cdot c0\right) \cdot \left(\color{blue}{0} + \frac{-1}{2} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot w\right)\right)}{{c0}^{2} \cdot {d}^{2}}\right)\right) \]

      7. +-lft-identity N/A

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

   5. Applied rewrites 66.6%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(-c0\right) \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{w \cdot h}{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}\right) \cdot -0.5\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(-c0\right) \cdot \left(\frac{\left(\left(M \cdot D\right) \cdot M\right) \cdot \left(\left(D \cdot w\right) \cdot h\right)}{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)} \cdot -0.5\right)\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 74.5%

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

   if -1.39999999999999993e-53 < c0

   1. Initial program 26.0%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 28.9

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

   5. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \left(c0 \cdot c0\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.1%

      \[\leadsto \left(d \cdot c0\right) \cdot \color{blue}{\frac{d \cdot c0}{\left(\left(w \cdot w\right) \cdot D\right) \cdot \left(D \cdot h\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.1%

      \[\leadsto \frac{c0 \cdot d}{\color{blue}{\left(\left(h \cdot \left(w \cdot D\right)\right) \cdot \frac{D}{c0 \cdot d}\right) \cdot w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -8 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(\frac{c0}{\left(\left(\left(D \cdot w\right) \cdot h\right) \cdot D\right) \cdot w} \cdot d\right) \cdot d\right) \cdot c0\\ \mathbf{elif}\;c0 \leq -1.4 \cdot 10^{-53}:\\ \;\;\;\;\left(\left(\left(--0.5\right) \cdot \left(\frac{\left(\left(h \cdot w\right) \cdot D\right) \cdot M}{d} \cdot \frac{M \cdot D}{\left(c0 \cdot c0\right) \cdot d}\right)\right) \cdot c0\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot c0}{\left(\frac{D}{d \cdot c0} \cdot \left(\left(D \cdot w\right) \cdot h\right)\right) \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ \mathbf{if}\;\left(\sqrt{t\_0 \cdot t\_0 - M \cdot M} + t\_0\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\left(\frac{c0}{w} \cdot d\right) \cdot \frac{d \cdot c0}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* (* d d) c0) (* (* D D) (* h w)))))
   (if (<=
        (* (+ (sqrt (- (* t_0 t_0) (* M M))) t_0) (/ c0 (* w 2.0)))
        INFINITY)
     (* (* (/ c0 w) d) (/ (* d c0) (* (* (* h w) D) D)))
     0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((D * D) * (h * w));
	double tmp;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= ((double) INFINITY)) {
		tmp = ((c0 / w) * d) * ((d * c0) / (((h * w) * D) * D));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) * c0) / ((D * D) * (h * w));
	double tmp;
	if (((Math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = ((c0 / w) * d) * ((d * c0) / (((h * w) * D) * D));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((d * d) * c0) / ((D * D) * (h * w))
	tmp = 0
	if ((math.sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= math.inf:
		tmp = ((c0 / w) * d) * ((d * c0) / (((h * w) * D) * D))
	else:
		tmp = 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) * c0) / Float64(Float64(D * D) * Float64(h * w)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))) + t_0) * Float64(c0 / Float64(w * 2.0))) <= Inf)
		tmp = Float64(Float64(Float64(c0 / w) * d) * Float64(Float64(d * c0) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((d * d) * c0) / ((D * D) * (h * w));
	tmp = 0.0;
	if (((sqrt(((t_0 * t_0) - (M * M))) + t_0) * (c0 / (w * 2.0))) <= Inf)
		tmp = ((c0 / w) * d) * ((d * c0) / (((h * w) * D) * D));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * N[(c0 / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(c0 / w), $MachinePrecision] * d), $MachinePrecision] * N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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))))) < +inf.0

   1. Initial program 73.6%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \color{blue}{\frac{{c0}^{2} \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)} \cdot \left(c0 \cdot c0\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 81.1%

      \[\leadsto \frac{d \cdot c0}{\left(D \cdot \left(h \cdot w\right)\right) \cdot D} \cdot \color{blue}{\frac{d \cdot c0}{w}} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.9%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) 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 0.0%

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

   3. Taylor expanded in c0 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{{c0}^{2} \cdot \left(-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}{w}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0} \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}\right) \]

      4. mul0-lft N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \frac{\color{blue}{0}}{w}\right) \]

      5. div0 N/A

         \[\leadsto \frac{-1}{2} \cdot \left({c0}^{2} \cdot \color{blue}{0}\right) \]

      6. mul0-rgt N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{0} \]

      7. metadata-eval 43.2

         \[\leadsto \color{blue}{0} \]

   5. Applied rewrites 43.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} - M \cdot M} + \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right) \cdot \frac{c0}{w \cdot 2} \leq \infty:\\ \;\;\;\;\left(\frac{c0}{w} \cdot d\right) \cdot \frac{d \cdot c0}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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))))))