Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.6% → 56.5%

Time: 28.2s

Alternatives: 13

Speedup: 151.0×

• Could not converge on a ground truth

  1. reg0 = (bf 37881931608525218986756879037373353260220000)
  2. reg1 = (bf "4.688419146052133278583754505736068797927e278")
  3. reg2 = (bf "-5.02543336362400688970024740712136834157e-197")
  4. reg3 = (bf "2.168700335487398331188895774878613202047e-170")
  5. reg4 = (bf "3.614945575298649741215610132872977829717e-66")
  6. reg5 = (bf #e4.971579495598485889103160396920270735873e77)

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: 24.6% 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: 56.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h}\\ t_1 := t\_0 \cdot \frac{d \cdot d}{D \cdot D}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_4 := t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M\_m \cdot M\_m}\right)\\ \mathbf{if}\;t\_4 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{w \cdot \frac{2}{c0}} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right)\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;{\left(\sqrt{2 \cdot t\_2} \cdot \left(\frac{d}{D} \cdot \sqrt{t\_0}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\log \left(e^{M\_m}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* w h)))
        (t_1 (* t_0 (/ (* d d) (* D D))))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_4 (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M_m M_m)))))))
   (if (<= t_4 2e+307)
     (* (/ 1.0 (* w (/ 2.0 c0))) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
     (if (<= t_4 INFINITY)
       (pow (* (sqrt (* 2.0 t_2)) (* (/ d D) (sqrt t_0))) 2.0)
       (* c0 (/ (log (exp M_m)) (* 2.0 w)))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))));
	double tmp;
	if (t_4 <= 2e+307) {
		tmp = (1.0 / (w * (2.0 / c0))) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = pow((sqrt((2.0 * t_2)) * ((d / D) * sqrt(t_0))), 2.0);
	} else {
		tmp = c0 * (log(exp(M_m)) / (2.0 * w));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (w * h);
	double t_1 = t_0 * ((d * d) / (D * D));
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_4 = t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M_m * M_m))));
	double tmp;
	if (t_4 <= 2e+307) {
		tmp = (1.0 / (w * (2.0 / c0))) * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))));
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.pow((Math.sqrt((2.0 * t_2)) * ((d / D) * Math.sqrt(t_0))), 2.0);
	} else {
		tmp = c0 * (Math.log(Math.exp(M_m)) / (2.0 * w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (w * h)
	t_1 = t_0 * ((d * d) / (D * D))
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_4 = t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M_m * M_m))))
	tmp = 0
	if t_4 <= 2e+307:
		tmp = (1.0 / (w * (2.0 / c0))) * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))
	elif t_4 <= math.inf:
		tmp = math.pow((math.sqrt((2.0 * t_2)) * ((d / D) * math.sqrt(t_0))), 2.0)
	else:
		tmp = c0 * (math.log(math.exp(M_m)) / (2.0 * w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(w * h))
	t_1 = Float64(t_0 * Float64(Float64(d * d) / Float64(D * D)))
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_4 = Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M_m * M_m)))))
	tmp = 0.0
	if (t_4 <= 2e+307)
		tmp = Float64(Float64(1.0 / Float64(w * Float64(2.0 / c0))) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m)))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * t_2)) * Float64(Float64(d / D) * sqrt(t_0))) ^ 2.0;
	else
		tmp = Float64(c0 * Float64(log(exp(M_m)) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (w * h);
	t_1 = t_0 * ((d * d) / (D * D));
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_4 = t_2 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))));
	tmp = 0.0;
	if (t_4 <= 2e+307)
		tmp = (1.0 / (w * (2.0 / c0))) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	elseif (t_4 <= Inf)
		tmp = (sqrt((2.0 * t_2)) * ((d / D) * sqrt(t_0))) ^ 2.0;
	else
		tmp = c0 * (log(exp(M_m)) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 2e+307], N[(N[(1.0 / N[(w * N[(2.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[Power[N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(c0 * N[(N[Log[N[Exp[M$95$m], $MachinePrecision]], $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))))) < 1.99999999999999997e307

   1. Initial program 82.8%

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

   3. Simplified 86.3%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. *-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

      2. associate-/l* 86.3%

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

   8. Simplified 86.3%

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

   if 1.99999999999999997e307 < (*.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 83.7%

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

   3. Simplified 83.5%

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

      1. frac-times 81.2%

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

      2. pow2 81.2%

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

      3. associate-*l* 81.2%

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

      4. pow2 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. associate-*r/ 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in c0 around inf 83.7%

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

       1. *-commutative 83.7%

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

       2. times-frac 83.5%

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

       3. unpow2 83.5%

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

       4. unpow2 83.5%

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

       5. times-frac 88.1%

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

       6. unpow2 88.1%

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

       7. *-commutative 88.1%

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

       8. associate-/r* 88.1%

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

   11. Simplified 88.1%

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

       1. add-sqr-sqrt 88.1%

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

       2. pow2 88.1%

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

       3. associate-*r* 88.1%

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

       4. sqrt-prod 88.1%

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

       5. *-commutative 88.1%

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

       6. sqrt-prod 88.1%

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

       7. sqrt-pow1 97.3%

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

       8. metadata-eval 97.3%

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

       9. pow1 97.3%

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

       10. associate-/l/ 97.4%

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

   13. Applied egg-rr 97.4%

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

   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%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 42.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{M}\right)}}{2 \cdot w} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{w \cdot \frac{2}{c0}} \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D} + \sqrt{\left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\right) - M \cdot M}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;{\left(\sqrt{2 \cdot \frac{c0}{2 \cdot w}} \cdot \left(\frac{d}{D} \cdot \sqrt{\frac{c0}{w \cdot h}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\log \left(e^{M}\right)}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \frac{d \cdot d}{D \cdot D}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right)\\ \mathbf{if}\;t\_3 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{w \cdot \frac{2}{c0}} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M\_m \cdot M\_m}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot {\left(\frac{d}{D} \cdot \sqrt{2 \cdot \frac{\frac{c0}{h}}{w}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\log \left(e^{M\_m}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (/ (* d d) (* D D))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m)))))))
   (if (<= t_3 2e+307)
     (* (/ 1.0 (* w (/ 2.0 c0))) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
     (if (<= t_3 INFINITY)
       (* t_1 (pow (* (/ d D) (sqrt (* 2.0 (/ (/ c0 h) w)))) 2.0))
       (* c0 (/ (log (exp M_m)) (* 2.0 w)))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))));
	double tmp;
	if (t_3 <= 2e+307) {
		tmp = (1.0 / (w * (2.0 / c0))) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * pow(((d / D) * sqrt((2.0 * ((c0 / h) / w)))), 2.0);
	} else {
		tmp = c0 * (log(exp(M_m)) / (2.0 * w));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M_m * M_m))));
	double tmp;
	if (t_3 <= 2e+307) {
		tmp = (1.0 / (w * (2.0 / c0))) * (t_0 + Math.sqrt(((t_0 * t_0) - (M_m * M_m))));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * Math.pow(((d / D) * Math.sqrt((2.0 * ((c0 / h) / w)))), 2.0);
	} else {
		tmp = c0 * (Math.log(Math.exp(M_m)) / (2.0 * w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M_m * M_m))))
	tmp = 0
	if t_3 <= 2e+307:
		tmp = (1.0 / (w * (2.0 / c0))) * (t_0 + math.sqrt(((t_0 * t_0) - (M_m * M_m))))
	elif t_3 <= math.inf:
		tmp = t_1 * math.pow(((d / D) * math.sqrt((2.0 * ((c0 / h) / w)))), 2.0)
	else:
		tmp = c0 * (math.log(math.exp(M_m)) / (2.0 * w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(d * d) / Float64(D * D)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m)))))
	tmp = 0.0
	if (t_3 <= 2e+307)
		tmp = Float64(Float64(1.0 / Float64(w * Float64(2.0 / c0))) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M_m * M_m)))));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * (Float64(Float64(d / D) * sqrt(Float64(2.0 * Float64(Float64(c0 / h) / w)))) ^ 2.0));
	else
		tmp = Float64(c0 * Float64(log(exp(M_m)) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (c0 / (w * h)) * ((d * d) / (D * D));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))));
	tmp = 0.0;
	if (t_3 <= 2e+307)
		tmp = (1.0 / (w * (2.0 / c0))) * (t_0 + sqrt(((t_0 * t_0) - (M_m * M_m))));
	elseif (t_3 <= Inf)
		tmp = t_1 * (((d / D) * sqrt((2.0 * ((c0 / h) / w)))) ^ 2.0);
	else
		tmp = c0 * (log(exp(M_m)) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e+307], N[(N[(1.0 / N[(w * N[(2.0 / c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[Power[N[(N[(d / D), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(c0 / h), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Log[N[Exp[M$95$m], $MachinePrecision]], $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))))) < 1.99999999999999997e307

   1. Initial program 82.8%

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

   3. Simplified 86.3%

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

      1. clear-num 86.3%

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

      2. inv-pow 86.3%

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

      3. *-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

      1. unpow-1 86.3%

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

      2. associate-/l* 86.3%

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

   8. Simplified 86.3%

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

   if 1.99999999999999997e307 < (*.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 83.7%

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

   3. Simplified 83.5%

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

      1. frac-times 81.2%

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

      2. pow2 81.2%

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

      3. associate-*l* 81.2%

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

      4. pow2 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. associate-*r/ 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in c0 around inf 83.7%

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

       1. *-commutative 83.7%

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

       2. times-frac 83.5%

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

       3. unpow2 83.5%

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

       4. unpow2 83.5%

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

       5. times-frac 88.1%

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

       6. unpow2 88.1%

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

       7. *-commutative 88.1%

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

       8. associate-/r* 88.1%

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

   11. Simplified 88.1%

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

       1. add-sqr-sqrt 88.1%

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

       2. pow2 88.1%

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

       3. associate-*r* 88.1%

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

       4. sqrt-prod 88.1%

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

       5. associate-/l/ 88.1%

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

       6. sqrt-pow1 95.0%

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

       7. metadata-eval 95.0%

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

       8. pow1 95.0%

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

   13. Applied egg-rr 95.0%

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

       1. *-commutative 95.0%

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

       2. associate-/r* 95.1%

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

   15. Simplified 95.1%

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

   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%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 42.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{M}\right)}}{2 \cdot w} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 56.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\log \left(e^{M\_m}\right)}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (* t_0 (* 2.0 (/ (* (/ c0 w) (pow (/ d D) 2.0)) h)))
     (* c0 (/ (log (exp M_m)) (* 2.0 w))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (((c0 / w) * pow((d / D), 2.0)) / h));
	} else {
		tmp = c0 * (log(exp(M_m)) / (2.0 * w));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * (((c0 / w) * Math.pow((d / D), 2.0)) / h));
	} else {
		tmp = c0 * (Math.log(Math.exp(M_m)) / (2.0 * w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = t_0 * (2.0 * (((c0 / w) * math.pow((d / D), 2.0)) / h))
	else:
		tmp = c0 * (math.log(math.exp(M_m)) / (2.0 * w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) / h)));
	else
		tmp = Float64(c0 * Float64(log(exp(M_m)) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = t_0 * (2.0 * (((c0 / w) * ((d / D) ^ 2.0)) / h));
	else
		tmp = c0 * (log(exp(M_m)) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Log[N[Exp[M$95$m], $MachinePrecision]], $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 83.2%

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

   3. Simplified 85.0%

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

      1. frac-times 80.1%

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

      2. pow2 80.1%

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

      3. associate-*l* 75.9%

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

      4. pow2 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-*r/ 78.0%

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

   8. Simplified 78.0%

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

   9. Taylor expanded in c0 around inf 81.2%

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

       1. *-commutative 81.2%

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

       2. times-frac 83.0%

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

       3. unpow2 83.0%

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

       4. unpow2 83.0%

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

       5. times-frac 85.1%

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

       6. unpow2 85.1%

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

       7. *-commutative 85.1%

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

       8. associate-/r* 85.1%

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

   11. Simplified 85.1%

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

       1. associate-*l/ 85.3%

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

   13. Applied egg-rr 85.3%

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

   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%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 42.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{M}\right)}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 54.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\left(M\_m + 1\right) + -1}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (* t_0 (* 2.0 (/ (* (/ c0 w) (pow (/ d D) 2.0)) h)))
     (* c0 (/ (+ (+ M_m 1.0) -1.0) (* 2.0 w))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * (2.0 * (((c0 / w) * pow((d / D), 2.0)) / h));
	} else {
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (2.0 * (((c0 / w) * Math.pow((d / D), 2.0)) / h));
	} else {
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = t_0 * (2.0 * (((c0 / w) * math.pow((d / D), 2.0)) / h))
	else:
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_0 * Float64(2.0 * Float64(Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) / h)));
	else
		tmp = Float64(c0 * Float64(Float64(Float64(M_m + 1.0) + -1.0) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = t_0 * (2.0 * (((c0 / w) * ((d / D) ^ 2.0)) / h));
	else
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[(N[(M$95$m + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

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 83.2%

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

   3. Simplified 85.0%

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

      1. frac-times 80.1%

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

      2. pow2 80.1%

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

      3. associate-*l* 75.9%

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

      4. pow2 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. associate-*r/ 78.0%

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

   8. Simplified 78.0%

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

   9. Taylor expanded in c0 around inf 81.2%

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

       1. *-commutative 81.2%

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

       2. times-frac 83.0%

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

       3. unpow2 83.0%

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

       4. unpow2 83.0%

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

       5. times-frac 85.1%

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

       6. unpow2 85.1%

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

       7. *-commutative 85.1%

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

       8. associate-/r* 85.1%

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

   11. Simplified 85.1%

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

       1. associate-*l/ 85.3%

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

   13. Applied egg-rr 85.3%

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

   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%

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

   3. Simplified 17.0%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 42.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{M}\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 40.6%

         \[\leadsto c0 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{M}\right)\right)\right)}}{2 \cdot w} \]

      2. rem-log-exp 22.1%

         \[\leadsto c0 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{M}\right)\right)}{2 \cdot w} \]

      3. log1p-define 41.9%

         \[\leadsto c0 \cdot \frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + M\right)}\right)}{2 \cdot w} \]

      4. expm1-define 41.9%

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

      5. add-exp-log 44.2%

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

      6. +-commutative 44.2%

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

   9. Applied egg-rr 44.2%

      \[\leadsto c0 \cdot \frac{\color{blue}{\left(M + 1\right) - 1}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\left(M + 1\right) + -1}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ \mathbf{if}\;M\_m \leq 2.25 \cdot 10^{-274}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(t\_0 \cdot \frac{d}{D \cdot \frac{D}{d}}\right)\right)\\ \mathbf{elif}\;M\_m \leq 3.5 \cdot 10^{-105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h)))
   (if (<= M_m 2.25e-274)
     (* (/ c0 (* 2.0 w)) (* 2.0 (* t_0 (/ d (* D (/ D d))))))
     (if (<= M_m 3.5e-105) 0.0 (* (* (/ c0 w) (pow (/ d D) 2.0)) t_0)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / w) / h;
	double tmp;
	if (M_m <= 2.25e-274) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * (d / (D * (D / d)))));
	} else if (M_m <= 3.5e-105) {
		tmp = 0.0;
	} else {
		tmp = ((c0 / w) * pow((d / D), 2.0)) * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c0 / w) / h
    if (m_m <= 2.25d-274) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (t_0 * (d_1 / (d * (d / d_1)))))
    else if (m_m <= 3.5d-105) then
        tmp = 0.0d0
    else
        tmp = ((c0 / w) * ((d_1 / d) ** 2.0d0)) * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / w) / h;
	double tmp;
	if (M_m <= 2.25e-274) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * (d / (D * (D / d)))));
	} else if (M_m <= 3.5e-105) {
		tmp = 0.0;
	} else {
		tmp = ((c0 / w) * Math.pow((d / D), 2.0)) * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = (c0 / w) / h
	tmp = 0
	if M_m <= 2.25e-274:
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * (d / (D * (D / d)))))
	elif M_m <= 3.5e-105:
		tmp = 0.0
	else:
		tmp = ((c0 / w) * math.pow((d / D), 2.0)) * t_0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 / w) / h)
	tmp = 0.0
	if (M_m <= 2.25e-274)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(t_0 * Float64(d / Float64(D * Float64(D / d))))));
	elseif (M_m <= 3.5e-105)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(c0 / w) * (Float64(d / D) ^ 2.0)) * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (c0 / w) / h;
	tmp = 0.0;
	if (M_m <= 2.25e-274)
		tmp = (c0 / (2.0 * w)) * (2.0 * (t_0 * (d / (D * (D / d)))));
	elseif (M_m <= 3.5e-105)
		tmp = 0.0;
	else
		tmp = ((c0 / w) * ((d / D) ^ 2.0)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[M$95$m, 2.25e-274], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(t$95$0 * N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 3.5e-105], 0.0, N[(N[(N[(c0 / w), $MachinePrecision] * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < 2.24999999999999996e-274

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

      1. frac-times 30.2%

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

      2. pow2 30.2%

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

      3. associate-*l* 28.7%

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

      4. pow2 28.7%

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

   6. Applied egg-rr 28.7%

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

      1. associate-*r/ 29.5%

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

   8. Simplified 29.5%

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

   9. Taylor expanded in c0 around inf 37.8%

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

       1. *-commutative 37.8%

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

       2. times-frac 37.0%

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

       3. unpow2 37.0%

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

       4. unpow2 37.0%

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

       5. times-frac 44.0%

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

       6. unpow2 44.0%

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

       7. *-commutative 44.0%

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

       8. associate-/r* 45.5%

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

   11. Simplified 45.5%

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

       1. unpow2 45.5%

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

       2. clear-num 45.5%

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

       3. frac-times 44.0%

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

       4. *-un-lft-identity 44.0%

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

   13. Applied egg-rr 44.0%

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

   if 2.24999999999999996e-274 < M < 3.5e-105

   1. Initial program 27.2%

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

   3. Simplified 32.3%

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

      1. frac-times 25.0%

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

      2. pow2 25.0%

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

      3. associate-*l* 17.4%

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

      4. pow2 17.4%

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

   6. Applied egg-rr 17.4%

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

      1. associate-*r/ 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in c0 around -inf 16.5%

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

       1. associate-*r* 16.5%

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

       2. mul-1-neg 16.5%

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

       3. distribute-lft1-in 16.5%

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

       4. metadata-eval 16.5%

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

       5. mul0-lft 62.7%

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

       6. distribute-lft-neg-in 62.7%

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

       7. distribute-rgt-neg-in 62.7%

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

       8. metadata-eval 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in c0 around 0 68.2%

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

   if 3.5e-105 < M

   1. Initial program 29.3%

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

   3. Simplified 29.3%

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

      1. frac-times 28.2%

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

      2. pow2 28.2%

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

      3. associate-*l* 29.3%

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

      4. pow2 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. associate-*r/ 29.3%

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

   8. Simplified 29.3%

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

   9. Taylor expanded in c0 around inf 42.0%

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

       1. *-commutative 42.0%

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

       2. times-frac 42.1%

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

       3. unpow2 42.1%

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

       4. unpow2 42.1%

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

       5. times-frac 51.5%

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

       6. unpow2 51.5%

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

       7. *-commutative 51.5%

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

       8. associate-/r* 53.7%

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

   11. Simplified 53.7%

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

       1. associate-*l/ 52.6%

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

       2. associate-/l/ 51.5%

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

   13. Applied egg-rr 51.5%

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

       1. associate-*r* 51.5%

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

       2. associate-*l/ 51.5%

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

       3. *-commutative 51.5%

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

       4. times-frac 51.5%

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

       5. metadata-eval 51.5%

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

       6. *-rgt-identity 51.5%

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

       7. rem-square-sqrt 30.9%

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

       8. *-rgt-identity 30.9%

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

       9. metadata-eval 30.9%

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

       10. times-frac 30.9%

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

       11. *-commutative 30.9%

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

       12. associate-*l/ 30.9%

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

       13. *-rgt-identity 30.9%

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

       14. metadata-eval 30.9%

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

       15. times-frac 30.9%

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

       16. *-commutative 30.9%

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

       17. associate-*l/ 30.9%

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

       18. *-commutative 30.9%

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

       19. unpow2 30.9%

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

       20. rem-square-sqrt 26.9%

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

   15. Simplified 54.8%

       \[\leadsto \color{blue}{\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \frac{\frac{c0}{w}}{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.25 \cdot 10^{-274}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{d}{D \cdot \frac{D}{d}}\right)\right)\\ \mathbf{elif}\;M \leq 3.5 \cdot 10^{-105}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c0}{w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot \frac{\frac{c0}{w}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 2.02 \cdot 10^{-274} \lor \neg \left(M\_m \leq 4.4 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (or (<= M_m 2.02e-274) (not (<= M_m 4.4e-106)))
   (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ (/ c0 w) h) (* (/ d D) (/ d D)))))
   0.0))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if ((M_m <= 2.02e-274) || !(M_m <= 4.4e-106)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if ((m_m <= 2.02d-274) .or. (.not. (m_m <= 4.4d-106))) then
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (((c0 / w) / h) * ((d_1 / d) * (d_1 / d))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if ((M_m <= 2.02e-274) || !(M_m <= 4.4e-106)) {
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if (M_m <= 2.02e-274) or not (M_m <= 4.4e-106):
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))))
	else:
		tmp = 0.0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if ((M_m <= 2.02e-274) || !(M_m <= 4.4e-106))
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(Float64(c0 / w) / h) * Float64(Float64(d / D) * Float64(d / D)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if ((M_m <= 2.02e-274) || ~((M_m <= 4.4e-106)))
		tmp = (c0 / (2.0 * w)) * (2.0 * (((c0 / w) / h) * ((d / D) * (d / D))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[Or[LessEqual[M$95$m, 2.02e-274], N[Not[LessEqual[M$95$m, 4.4e-106]], $MachinePrecision]], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if M < 2.01999999999999993e-274 or 4.39999999999999989e-106 < M

   1. Initial program 30.4%

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

   3. Simplified 30.3%

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

      1. frac-times 29.4%

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

      2. pow2 29.4%

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

      3. associate-*l* 28.9%

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

      4. pow2 28.9%

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

   6. Applied egg-rr 28.9%

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

      1. associate-*r/ 29.4%

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

   8. Simplified 29.4%

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

   9. Taylor expanded in c0 around inf 39.5%

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

       1. *-commutative 39.5%

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

       2. times-frac 39.0%

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

       3. unpow2 39.0%

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

       4. unpow2 39.0%

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

       5. times-frac 46.9%

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

       6. unpow2 46.9%

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

       7. *-commutative 46.9%

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

       8. associate-/r* 48.7%

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

   11. Simplified 48.7%

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

       1. unpow2 48.7%

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

   13. Applied egg-rr 48.7%

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

   if 2.01999999999999993e-274 < M < 4.39999999999999989e-106

   1. Initial program 27.2%

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

   3. Simplified 32.3%

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

      1. frac-times 25.0%

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

      2. pow2 25.0%

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

      3. associate-*l* 17.4%

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

      4. pow2 17.4%

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

   6. Applied egg-rr 17.4%

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

      1. associate-*r/ 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in c0 around -inf 16.5%

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

       1. associate-*r* 16.5%

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

       2. mul-1-neg 16.5%

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

       3. distribute-lft1-in 16.5%

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

       4. metadata-eval 16.5%

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

       5. mul0-lft 62.7%

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

       6. distribute-lft-neg-in 62.7%

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

       7. distribute-rgt-neg-in 62.7%

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

       8. metadata-eval 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in c0 around 0 68.2%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.02 \cdot 10^{-274} \lor \neg \left(M \leq 4.4 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{\frac{c0}{w}}{h}\\ t_1 := \frac{c0}{2 \cdot w}\\ \mathbf{if}\;M\_m \leq 2.4 \cdot 10^{-274}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_0 \cdot \frac{d}{D \cdot \frac{D}{d}}\right)\right)\\ \mathbf{elif}\;M\_m \leq 9.6 \cdot 10^{-107}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(2 \cdot \left(t\_0 \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ (/ c0 w) h)) (t_1 (/ c0 (* 2.0 w))))
   (if (<= M_m 2.4e-274)
     (* t_1 (* 2.0 (* t_0 (/ d (* D (/ D d))))))
     (if (<= M_m 9.6e-107) 0.0 (* t_1 (* 2.0 (* t_0 (* (/ d D) (/ d D)))))))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / w) / h;
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (M_m <= 2.4e-274) {
		tmp = t_1 * (2.0 * (t_0 * (d / (D * (D / d)))));
	} else if (M_m <= 9.6e-107) {
		tmp = 0.0;
	} else {
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c0 / w) / h
    t_1 = c0 / (2.0d0 * w)
    if (m_m <= 2.4d-274) then
        tmp = t_1 * (2.0d0 * (t_0 * (d_1 / (d * (d / d_1)))))
    else if (m_m <= 9.6d-107) then
        tmp = 0.0d0
    else
        tmp = t_1 * (2.0d0 * (t_0 * ((d_1 / d) * (d_1 / d))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = (c0 / w) / h;
	double t_1 = c0 / (2.0 * w);
	double tmp;
	if (M_m <= 2.4e-274) {
		tmp = t_1 * (2.0 * (t_0 * (d / (D * (D / d)))));
	} else if (M_m <= 9.6e-107) {
		tmp = 0.0;
	} else {
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = (c0 / w) / h
	t_1 = c0 / (2.0 * w)
	tmp = 0
	if M_m <= 2.4e-274:
		tmp = t_1 * (2.0 * (t_0 * (d / (D * (D / d)))))
	elif M_m <= 9.6e-107:
		tmp = 0.0
	else:
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	t_0 = Float64(Float64(c0 / w) / h)
	t_1 = Float64(c0 / Float64(2.0 * w))
	tmp = 0.0
	if (M_m <= 2.4e-274)
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(d / Float64(D * Float64(D / d))))));
	elseif (M_m <= 9.6e-107)
		tmp = 0.0;
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(t_0 * Float64(Float64(d / D) * Float64(d / D)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	t_0 = (c0 / w) / h;
	t_1 = c0 / (2.0 * w);
	tmp = 0.0;
	if (M_m <= 2.4e-274)
		tmp = t_1 * (2.0 * (t_0 * (d / (D * (D / d)))));
	elseif (M_m <= 9.6e-107)
		tmp = 0.0;
	else
		tmp = t_1 * (2.0 * (t_0 * ((d / D) * (d / D))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 / w), $MachinePrecision] / h), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M$95$m, 2.4e-274], N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(d / N[(D * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[M$95$m, 9.6e-107], 0.0, N[(t$95$1 * N[(2.0 * N[(t$95$0 * N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if M < 2.4e-274

   1. Initial program 31.0%

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

   3. Simplified 31.0%

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

      1. frac-times 30.2%

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

      2. pow2 30.2%

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

      3. associate-*l* 28.7%

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

      4. pow2 28.7%

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

   6. Applied egg-rr 28.7%

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

      1. associate-*r/ 29.5%

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

   8. Simplified 29.5%

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

   9. Taylor expanded in c0 around inf 37.8%

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

       1. *-commutative 37.8%

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

       2. times-frac 37.0%

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

       3. unpow2 37.0%

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

       4. unpow2 37.0%

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

       5. times-frac 44.0%

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

       6. unpow2 44.0%

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

       7. *-commutative 44.0%

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

       8. associate-/r* 45.5%

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

   11. Simplified 45.5%

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

       1. unpow2 45.5%

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

       2. clear-num 45.5%

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

       3. frac-times 44.0%

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

       4. *-un-lft-identity 44.0%

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

   13. Applied egg-rr 44.0%

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

   if 2.4e-274 < M < 9.59999999999999977e-107

   1. Initial program 27.2%

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

   3. Simplified 32.3%

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

      1. frac-times 25.0%

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

      2. pow2 25.0%

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

      3. associate-*l* 17.4%

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

      4. pow2 17.4%

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

   6. Applied egg-rr 17.4%

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

      1. associate-*r/ 20.1%

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

   8. Simplified 20.1%

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

   9. Taylor expanded in c0 around -inf 16.5%

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

       1. associate-*r* 16.5%

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

       2. mul-1-neg 16.5%

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

       3. distribute-lft1-in 16.5%

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

       4. metadata-eval 16.5%

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

       5. mul0-lft 62.7%

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

       6. distribute-lft-neg-in 62.7%

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

       7. distribute-rgt-neg-in 62.7%

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

       8. metadata-eval 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in c0 around 0 68.2%

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

   if 9.59999999999999977e-107 < M

   1. Initial program 29.3%

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

   3. Simplified 29.3%

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

      1. frac-times 28.2%

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

      2. pow2 28.2%

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

      3. associate-*l* 29.3%

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

      4. pow2 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. associate-*r/ 29.3%

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

   8. Simplified 29.3%

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

   9. Taylor expanded in c0 around inf 42.0%

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

       1. *-commutative 42.0%

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

       2. times-frac 42.1%

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

       3. unpow2 42.1%

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

       4. unpow2 42.1%

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

       5. times-frac 51.5%

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

       6. unpow2 51.5%

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

       7. *-commutative 51.5%

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

       8. associate-/r* 53.7%

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

   11. Simplified 53.7%

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

       1. unpow2 53.7%

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

   13. Applied egg-rr 53.7%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \color{blue}{\left(\frac{d}{D} \cdot \frac{d}{D}\right)}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 2.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \frac{d}{D \cdot \frac{D}{d}}\right)\right)\\ \mathbf{elif}\;M \leq 9.6 \cdot 10^{-107}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{\frac{c0}{w}}{h} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 4.1 \cdot 10^{-25}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{M\_m \cdot \left(2 + M\_m\right)}{M\_m}}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 4.1e-25) 0.0 (* c0 (/ (/ (* M_m (+ 2.0 M_m)) M_m) (* 2.0 w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 4.1e-25) {
		tmp = 0.0;
	} else {
		tmp = c0 * (((M_m * (2.0 + M_m)) / M_m) / (2.0 * w));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 4.1d-25) then
        tmp = 0.0d0
    else
        tmp = c0 * (((m_m * (2.0d0 + m_m)) / m_m) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 4.1e-25) {
		tmp = 0.0;
	} else {
		tmp = c0 * (((M_m * (2.0 + M_m)) / M_m) / (2.0 * w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 4.1e-25:
		tmp = 0.0
	else:
		tmp = c0 * (((M_m * (2.0 + M_m)) / M_m) / (2.0 * w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 4.1e-25)
		tmp = 0.0;
	else
		tmp = Float64(c0 * Float64(Float64(Float64(M_m * Float64(2.0 + M_m)) / M_m) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 4.1e-25)
		tmp = 0.0;
	else
		tmp = c0 * (((M_m * (2.0 + M_m)) / M_m) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 4.1e-25], 0.0, N[(c0 * N[(N[(N[(M$95$m * N[(2.0 + M$95$m), $MachinePrecision]), $MachinePrecision] / M$95$m), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 4.09999999999999987e-25

   1. Initial program 30.8%

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

   3. Simplified 31.7%

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

      1. frac-times 29.3%

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

      2. pow2 29.3%

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

      3. associate-*l* 27.3%

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

      4. pow2 27.3%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*r/ 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in c0 around -inf 5.4%

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

       1. associate-*r* 5.4%

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

       2. mul-1-neg 5.4%

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

       3. distribute-lft1-in 5.4%

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

       4. metadata-eval 5.4%

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

       5. mul0-lft 34.4%

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

       6. distribute-lft-neg-in 34.4%

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

       7. distribute-rgt-neg-in 34.4%

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

       8. metadata-eval 34.4%

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

   11. Simplified 34.4%

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

   12. Taylor expanded in c0 around 0 39.8%

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

   if 4.09999999999999987e-25 < M

   1. Initial program 27.2%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 36.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M\right)} - -1}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. sub-neg 36.9%

         \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M\right)} + \left(--1\right)}}{2 \cdot w} \]

      2. log1p-undefine 36.9%

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

      3. rem-exp-log 36.9%

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

      4. metadata-eval 36.9%

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

   9. Simplified 36.9%

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

       1. flip-+ 35.2%

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

       2. pow2 35.2%

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

       3. +-commutative 35.2%

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

       4. metadata-eval 35.2%

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

       5. add-exp-log 35.2%

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

       6. expm1-define 35.2%

          \[\leadsto c0 \cdot \frac{\frac{{\left(M + 1\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(\log \left(1 + M\right)\right)}}}{2 \cdot w} \]

       7. log1p-define 35.3%

          \[\leadsto c0 \cdot \frac{\frac{{\left(M + 1\right)}^{2} - 1}{\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(M\right)}\right)}}{2 \cdot w} \]

       8. expm1-log1p-u 35.3%

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

   11. Applied egg-rr 35.3%

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

       1. unpow2 35.3%

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

       2. difference-of-sqr-1 35.3%

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

       3. associate-+l+ 35.3%

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

       4. metadata-eval 35.3%

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

       5. associate--l+ 36.9%

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

       6. metadata-eval 36.9%

          \[\leadsto c0 \cdot \frac{\frac{\left(M + 2\right) \cdot \left(M + \color{blue}{0}\right)}{M}}{2 \cdot w} \]

       7. +-rgt-identity 36.9%

          \[\leadsto c0 \cdot \frac{\frac{\left(M + 2\right) \cdot \color{blue}{M}}{M}}{2 \cdot w} \]

   13. Simplified 36.9%

       \[\leadsto c0 \cdot \frac{\color{blue}{\frac{\left(M + 2\right) \cdot M}{M}}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 4.1 \cdot 10^{-25}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{M \cdot \left(2 + M\right)}{M}}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 37.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;w \leq -1.08 \cdot 10^{+60}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\left(M\_m + 1\right) + -1}{2 \cdot w}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= w -1.08e+60) 0.0 (* c0 (/ (+ (+ M_m 1.0) -1.0) (* 2.0 w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (w <= -1.08e+60) {
		tmp = 0.0;
	} else {
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (w <= (-1.08d+60)) then
        tmp = 0.0d0
    else
        tmp = c0 * (((m_m + 1.0d0) + (-1.0d0)) / (2.0d0 * w))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (w <= -1.08e+60) {
		tmp = 0.0;
	} else {
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if w <= -1.08e+60:
		tmp = 0.0
	else:
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (w <= -1.08e+60)
		tmp = 0.0;
	else
		tmp = Float64(c0 * Float64(Float64(Float64(M_m + 1.0) + -1.0) / Float64(2.0 * w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (w <= -1.08e+60)
		tmp = 0.0;
	else
		tmp = c0 * (((M_m + 1.0) + -1.0) / (2.0 * w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[w, -1.08e+60], 0.0, N[(c0 * N[(N[(N[(M$95$m + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.08e60

   1. Initial program 23.5%

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

   3. Simplified 25.8%

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

      1. frac-times 21.3%

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

      2. pow2 21.3%

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

      3. associate-*l* 21.2%

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

      4. pow2 21.2%

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

   6. Applied egg-rr 21.2%

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

      1. associate-*r/ 23.5%

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

   8. Simplified 23.5%

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

   9. Taylor expanded in c0 around -inf 11.9%

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

       1. associate-*r* 11.9%

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

       2. mul-1-neg 11.9%

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

       3. distribute-lft1-in 11.9%

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

       4. metadata-eval 11.9%

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

       5. mul0-lft 50.5%

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

       6. distribute-lft-neg-in 50.5%

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

       7. distribute-rgt-neg-in 50.5%

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

       8. metadata-eval 50.5%

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

   11. Simplified 50.5%

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

   12. Taylor expanded in c0 around 0 50.5%

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

   if -1.08e60 < w

   1. Initial program 31.2%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 36.4%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{M}\right)}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 35.2%

         \[\leadsto c0 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{M}\right)\right)\right)}}{2 \cdot w} \]

      2. rem-log-exp 18.3%

         \[\leadsto c0 \cdot \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{M}\right)\right)}{2 \cdot w} \]

      3. log1p-define 33.6%

         \[\leadsto c0 \cdot \frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + M\right)}\right)}{2 \cdot w} \]

      4. expm1-define 33.6%

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

      5. add-exp-log 34.9%

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

      6. +-commutative 34.9%

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

   9. Applied egg-rr 34.9%

      \[\leadsto c0 \cdot \frac{\color{blue}{\left(M + 1\right) - 1}}{2 \cdot w} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.08 \cdot 10^{+60}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\left(M + 1\right) + -1}{2 \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9.5 \cdot 10^{-24}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\frac{M\_m}{w} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 9.5e-24) 0.0 (* c0 (* (/ M_m w) 0.5))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.5e-24) {
		tmp = 0.0;
	} else {
		tmp = c0 * ((M_m / w) * 0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 9.5d-24) then
        tmp = 0.0d0
    else
        tmp = c0 * ((m_m / w) * 0.5d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.5e-24) {
		tmp = 0.0;
	} else {
		tmp = c0 * ((M_m / w) * 0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 9.5e-24:
		tmp = 0.0
	else:
		tmp = c0 * ((M_m / w) * 0.5)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 9.5e-24)
		tmp = 0.0;
	else
		tmp = Float64(c0 * Float64(Float64(M_m / w) * 0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 9.5e-24)
		tmp = 0.0;
	else
		tmp = c0 * ((M_m / w) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 9.5e-24], 0.0, N[(c0 * N[(N[(M$95$m / w), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.50000000000000029e-24

   1. Initial program 30.8%

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

   3. Simplified 31.7%

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

      1. frac-times 29.3%

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

      2. pow2 29.3%

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

      3. associate-*l* 27.3%

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

      4. pow2 27.3%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*r/ 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in c0 around -inf 5.4%

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

       1. associate-*r* 5.4%

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

       2. mul-1-neg 5.4%

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

       3. distribute-lft1-in 5.4%

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

       4. metadata-eval 5.4%

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

       5. mul0-lft 34.4%

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

       6. distribute-lft-neg-in 34.4%

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

       7. distribute-rgt-neg-in 34.4%

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

       8. metadata-eval 34.4%

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

   11. Simplified 34.4%

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

   12. Taylor expanded in c0 around 0 39.8%

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

   if 9.50000000000000029e-24 < M

   1. Initial program 27.2%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 36.9%

      \[\leadsto c0 \cdot \color{blue}{\left(\frac{M}{w} \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 36.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9.5 \cdot 10^{-24}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(M\_m \cdot \frac{0.5}{w}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 9.5e-24) 0.0 (* c0 (* M_m (/ 0.5 w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.5e-24) {
		tmp = 0.0;
	} else {
		tmp = c0 * (M_m * (0.5 / w));
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (m_m <= 9.5d-24) then
        tmp = 0.0d0
    else
        tmp = c0 * (m_m * (0.5d0 / w))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.5e-24) {
		tmp = 0.0;
	} else {
		tmp = c0 * (M_m * (0.5 / w));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 9.5e-24:
		tmp = 0.0
	else:
		tmp = c0 * (M_m * (0.5 / w))
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 9.5e-24)
		tmp = 0.0;
	else
		tmp = Float64(c0 * Float64(M_m * Float64(0.5 / w)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 9.5e-24)
		tmp = 0.0;
	else
		tmp = c0 * (M_m * (0.5 / w));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 9.5e-24], 0.0, N[(c0 * N[(M$95$m * N[(0.5 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.50000000000000029e-24

   1. Initial program 30.8%

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

   3. Simplified 31.7%

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

      1. frac-times 29.3%

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

      2. pow2 29.3%

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

      3. associate-*l* 27.3%

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

      4. pow2 27.3%

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

   6. Applied egg-rr 27.3%

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

      1. associate-*r/ 28.3%

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

   8. Simplified 28.3%

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

   9. Taylor expanded in c0 around -inf 5.4%

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

       1. associate-*r* 5.4%

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

       2. mul-1-neg 5.4%

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

       3. distribute-lft1-in 5.4%

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

       4. metadata-eval 5.4%

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

       5. mul0-lft 34.4%

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

       6. distribute-lft-neg-in 34.4%

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

       7. distribute-rgt-neg-in 34.4%

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

       8. metadata-eval 34.4%

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

   11. Simplified 34.4%

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

   12. Taylor expanded in c0 around 0 39.8%

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

   if 9.50000000000000029e-24 < M

   1. Initial program 27.2%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 40.7%

      \[\leadsto c0 \cdot \frac{\color{blue}{\log \left(e^{M}\right)}}{2 \cdot w} \]

   8. Taylor expanded in M around 0 36.9%

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

      1. associate-*r/ 36.9%

         \[\leadsto c0 \cdot \color{blue}{\frac{0.5 \cdot M}{w}} \]

      2. *-commutative 36.9%

         \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot 0.5}}{w} \]

      3. associate-*r/ 36.9%

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

   10. Simplified 36.9%

       \[\leadsto c0 \cdot \color{blue}{\left(M \cdot \frac{0.5}{w}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 33.6% accurate, 18.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;c0 \leq -6.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{c0}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= c0 -6.4e+168) (/ c0 w) 0.0))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (c0 <= -6.4e+168) {
		tmp = c0 / w;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    real(8) :: tmp
    if (c0 <= (-6.4d+168)) then
        tmp = c0 / w
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (c0 <= -6.4e+168) {
		tmp = c0 / w;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if c0 <= -6.4e+168:
		tmp = c0 / w
	else:
		tmp = 0.0
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (c0 <= -6.4e+168)
		tmp = Float64(c0 / w);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (c0 <= -6.4e+168)
		tmp = c0 / w;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[c0, -6.4e+168], N[(c0 / w), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if c0 < -6.4000000000000002e168

   1. Initial program 41.3%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in c0 around 0 0.0%

      \[\leadsto c0 \cdot \frac{\color{blue}{M \cdot \sqrt{-1}}}{2 \cdot w} \]

   7. Applied egg-rr 28.3%

      \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M\right)} - -1}}{2 \cdot w} \]
   8. Step-by-step derivation

      1. sub-neg 28.3%

         \[\leadsto c0 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(M\right)} + \left(--1\right)}}{2 \cdot w} \]

      2. log1p-undefine 28.3%

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

      3. rem-exp-log 29.2%

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

      4. metadata-eval 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in M around 0 31.6%

       \[\leadsto \color{blue}{\frac{c0}{w}} \]

   if -6.4000000000000002e168 < c0

   1. Initial program 28.2%

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

   3. Simplified 28.6%

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

      1. frac-times 26.8%

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

      2. pow2 26.8%

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

      3. associate-*l* 26.0%

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

      4. pow2 26.0%

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

   6. Applied egg-rr 26.0%

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

      1. associate-*r/ 26.9%

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

   8. Simplified 26.9%

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

   9. Taylor expanded in c0 around -inf 4.7%

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

       1. associate-*r* 4.7%

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

       2. mul-1-neg 4.7%

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

       3. distribute-lft1-in 4.7%

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

       4. metadata-eval 4.7%

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

       5. mul0-lft 32.9%

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

       6. distribute-lft-neg-in 32.9%

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

       7. distribute-rgt-neg-in 32.9%

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

       8. metadata-eval 32.9%

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

   11. Simplified 32.9%

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

   12. Taylor expanded in c0 around 0 37.2%

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

Alternative 13: 34.1% accurate, 151.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 0.0)

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Fortran

M_m = abs(m)
real(8) function code(c0, w, h, d, d_1, m_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_m
    code = 0.0d0
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return 0.0;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return 0.0

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	return 0.0
end

MATLAB

M_m = abs(M);
function tmp = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := 0.0

Derivation

1. Initial program 29.9%

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

3. Simplified 30.6%

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

   1. frac-times 28.8%

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

   2. pow2 28.8%

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

   3. associate-*l* 27.3%

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

   4. pow2 27.3%

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

6. Applied egg-rr 27.3%

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

   1. associate-*r/ 28.1%

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

8. Simplified 28.1%

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

9. Taylor expanded in c0 around -inf 4.1%

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

    1. associate-*r* 4.1%

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

    2. mul-1-neg 4.1%

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

    3. distribute-lft1-in 4.1%

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

    4. metadata-eval 4.1%

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

    5. mul0-lft 29.9%

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

    6. distribute-lft-neg-in 29.9%

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

    7. distribute-rgt-neg-in 29.9%

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

    8. metadata-eval 29.9%

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

11. Simplified 29.9%

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

12. Taylor expanded in c0 around 0 34.1%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))))))