Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 49.3%

Time: 46.2s

Alternatives: 11

Speedup: 30.2×

• could not determine a ground truth

  1. c0 = -4.143387550884428e+263
  2. w = -4.980674036810121e-263
  3. h = -9.707583621765186e-11
  4. D = -8.604663570142654e-301
  5. d = 1.6510682270793985e+250
  6. M = -1.8528260960251538e-224

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: 24.3% 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: 49.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ t_1 := \frac{c0}{2 \cdot w} \cdot \left(t_0 + \sqrt{t_0 \cdot t_0 - M \cdot M}\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h))))
        (t_1 (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 (/ 0.0 (* w h)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.0 / (w * h);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	double t_1 = (c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.0 / (w * h);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	t_1 = (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.0 / (w * h)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 78.0%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 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 1.1%

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

   4. Taylor expanded in c0 around -inf 0.1%

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

      1. associate-*r/ 0.1%

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

      2. unpow2 0.1%

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

      3. neg-mul-1 0.1%

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

      4. associate-*r* 0.2%

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

      5. unpow2 0.2%

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

   6. Simplified 0.2%

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

   7. Taylor expanded in h around 0 1.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 1.3%

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

   9. Simplified 38.4%

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

4. Final simplification 51.7%

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

Alternative 2: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.8 \cdot 10^{+67} \lor \neg \left(w \leq -1.2 \cdot 10^{-110} \lor \neg \left(w \leq -1.15 \cdot 10^{-165}\right) \land w \leq 6.8 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= w -4.8e+67)
         (not
          (or (<= w -1.2e-110) (and (not (<= w -1.15e-165)) (<= w 6.8e+20)))))
   (/ 0.0 (* w h))
   (* (/ c0 (* w h)) (* (/ c0 (* 2.0 w)) (* 2.0 (pow (/ d D) 2.0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -4.8e+67) || !((w <= -1.2e-110) || (!(w <= -1.15e-165) && (w <= 6.8e+20)))) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = (c0 / (w * h)) * ((c0 / (2.0 * w)) * (2.0 * pow((d / D), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((w <= (-4.8d+67)) .or. (.not. (w <= (-1.2d-110)) .or. (.not. (w <= (-1.15d-165))) .and. (w <= 6.8d+20))) then
        tmp = 0.0d0 / (w * h)
    else
        tmp = (c0 / (w * h)) * ((c0 / (2.0d0 * w)) * (2.0d0 * ((d_1 / d) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -4.8e+67) || !((w <= -1.2e-110) || (!(w <= -1.15e-165) && (w <= 6.8e+20)))) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = (c0 / (w * h)) * ((c0 / (2.0 * w)) * (2.0 * Math.pow((d / D), 2.0)));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (w <= -4.8e+67) or not ((w <= -1.2e-110) or (not (w <= -1.15e-165) and (w <= 6.8e+20))):
		tmp = 0.0 / (w * h)
	else:
		tmp = (c0 / (w * h)) * ((c0 / (2.0 * w)) * (2.0 * math.pow((d / D), 2.0)))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((w <= -4.8e+67) || !((w <= -1.2e-110) || (!(w <= -1.15e-165) && (w <= 6.8e+20))))
		tmp = Float64(0.0 / Float64(w * h));
	else
		tmp = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * (Float64(d / D) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((w <= -4.8e+67) || ~(((w <= -1.2e-110) || (~((w <= -1.15e-165)) && (w <= 6.8e+20)))))
		tmp = 0.0 / (w * h);
	else
		tmp = (c0 / (w * h)) * ((c0 / (2.0 * w)) * (2.0 * ((d / D) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[w, -4.8e+67], N[Not[Or[LessEqual[w, -1.2e-110], And[N[Not[LessEqual[w, -1.15e-165]], $MachinePrecision], LessEqual[w, 6.8e+20]]]], $MachinePrecision]], N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4.80000000000000004e67 or -1.20000000000000003e-110 < w < -1.15e-165 or 6.8e20 < w

   1. Initial program 18.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in c0 around -inf 4.1%

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

      1. associate-*r/ 4.1%

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

      2. unpow2 4.1%

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

      3. neg-mul-1 4.1%

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

      4. associate-*r* 5.0%

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

      5. unpow2 5.0%

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

   6. Simplified 5.0%

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

   7. Taylor expanded in h around 0 9.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 9.0%

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

   9. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]

   if -4.80000000000000004e67 < w < -1.20000000000000003e-110 or -1.15e-165 < w < 6.8e20

   1. Initial program 31.5%

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

   3. Simplified 40.1%

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

   4. Taylor expanded in D around 0 9.1%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{0.5 \cdot \frac{h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}\right)\right)}{c0 \cdot {d}^{2}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]
   5. Step-by-step derivation

      1. fma-def 9.1%

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

   6. Simplified 10.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right)} \]

      2. associate-/r* 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{h \cdot w}} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      3. pow2 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{h \cdot w} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      4. *-commutative 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      5. *-commutative 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{\color{blue}{w \cdot h}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

   8. Applied egg-rr 44.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + \mathsf{fma}\left(0.5, \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right) + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)} \]

      2. fma-udef 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right) \]

      3. associate-+l+ 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)} + \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)} \]

   10. Simplified 46.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{h}{c0} \cdot 0 + \left(\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot 2\right)} \]
   11. Step-by-step derivation

       1. mul0-rgt 48.3%

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

       2. distribute-lft-in 34.1%

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

       3. *-commutative 34.1%

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

       4. *-commutative 34.1%

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

       5. pow2 34.1%

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

       6. associate-*l* 34.1%

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

       7. pow2 34.1%

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

   12. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\frac{c0}{w \cdot 2} \cdot 0 + \frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot 2\right)\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 34.1%

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

       2. distribute-lft-in 48.3%

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

       3. +-lft-identity 48.3%

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

       4. *-commutative 48.3%

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

       5. +-lft-identity 48.3%

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

       6. associate-*l* 48.3%

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

       7. *-commutative 48.3%

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

       8. *-commutative 48.3%

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

   14. Simplified 48.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.8 \cdot 10^{+67} \lor \neg \left(w \leq -1.2 \cdot 10^{-110} \lor \neg \left(w \leq -1.15 \cdot 10^{-165}\right) \land w \leq 6.8 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 3: 43.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w \cdot h} \cdot \left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ t_1 := \frac{0}{w \cdot h}\\ \mathbf{if}\;w \leq -9.2 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \leq -6.4 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -7 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}\right)\\ \mathbf{elif}\;w \leq 3.25 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ c0 (* w h)) (* (/ c0 (* 2.0 w)) (* 2.0 (pow (/ d D) 2.0)))))
        (t_1 (/ 0.0 (* w h))))
   (if (<= w -9.2e+67)
     t_1
     (if (<= w -6.4e-114)
       t_0
       (if (<= w -7e-174)
         (fma (* c0 c0) 0.0 (/ (* 0.25 (* (* D D) (* h (* M M)))) (* d d)))
         (if (<= w 3.25e+25) t_0 t_1))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 / (w * h)) * ((c0 / (2.0 * w)) * (2.0 * pow((d / D), 2.0)));
	double t_1 = 0.0 / (w * h);
	double tmp;
	if (w <= -9.2e+67) {
		tmp = t_1;
	} else if (w <= -6.4e-114) {
		tmp = t_0;
	} else if (w <= -7e-174) {
		tmp = fma((c0 * c0), 0.0, ((0.25 * ((D * D) * (h * (M * M)))) / (d * d)));
	} else if (w <= 3.25e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 / Float64(w * h)) * Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * (Float64(d / D) ^ 2.0))))
	t_1 = Float64(0.0 / Float64(w * h))
	tmp = 0.0
	if (w <= -9.2e+67)
		tmp = t_1;
	elseif (w <= -6.4e-114)
		tmp = t_0;
	elseif (w <= -7e-174)
		tmp = fma(Float64(c0 * c0), 0.0, Float64(Float64(0.25 * Float64(Float64(D * D) * Float64(h * Float64(M * M)))) / Float64(d * d)));
	elseif (w <= 3.25e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -9.2e+67], t$95$1, If[LessEqual[w, -6.4e-114], t$95$0, If[LessEqual[w, -7e-174], N[(N[(c0 * c0), $MachinePrecision] * 0.0 + N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 3.25e+25], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -9.1999999999999994e67 or 3.25000000000000003e25 < w

   1. Initial program 18.0%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in c0 around -inf 4.7%

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

      1. associate-*r/ 4.7%

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

      2. unpow2 4.7%

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

      3. neg-mul-1 4.7%

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

      4. associate-*r* 5.7%

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

      5. unpow2 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in h around 0 10.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 10.3%

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

   9. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]

   if -9.1999999999999994e67 < w < -6.4000000000000003e-114 or -6.99999999999999975e-174 < w < 3.25000000000000003e25

   1. Initial program 31.5%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in D around 0 9.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{0.5 \cdot \frac{h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}\right)\right)}{c0 \cdot {d}^{2}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]
   5. Step-by-step derivation

      1. fma-def 9.0%

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

   6. Simplified 10.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right)} \]

      2. associate-/r* 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{h \cdot w}} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      3. pow2 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{h \cdot w} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      4. *-commutative 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      5. *-commutative 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{\color{blue}{w \cdot h}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

   8. Applied egg-rr 44.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + \mathsf{fma}\left(0.5, \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right) + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)} \]

      2. fma-udef 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right) \]

      3. associate-+l+ 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)} + \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)} \]

   10. Simplified 46.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{h}{c0} \cdot 0 + \left(\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot 2\right)} \]
   11. Step-by-step derivation

       1. mul0-rgt 48.3%

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

       2. distribute-lft-in 34.0%

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

       3. *-commutative 34.0%

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

       4. *-commutative 34.0%

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

       5. pow2 34.0%

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

       6. associate-*l* 34.0%

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

       7. pow2 34.0%

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

   12. Applied egg-rr 34.0%

       \[\leadsto \color{blue}{\frac{c0}{w \cdot 2} \cdot 0 + \frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot 2\right)\right)} \]
   13. Step-by-step derivation

       1. +-lft-identity 34.0%

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

       2. distribute-lft-in 48.3%

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

       3. +-lft-identity 48.3%

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

       4. *-commutative 48.3%

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

       5. +-lft-identity 48.3%

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

       6. associate-*l* 48.3%

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

       7. *-commutative 48.3%

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

       8. *-commutative 48.3%

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

   14. Simplified 48.3%

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

   if -6.4000000000000003e-114 < w < -6.99999999999999975e-174

   1. Initial program 21.8%

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

      1. distribute-lft-in 21.8%

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

      2. *-commutative 21.8%

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

      3. times-frac 21.5%

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

      4. frac-times 22.0%

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

      5. pow2 22.0%

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

      6. *-commutative 22.0%

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

   3. Applied egg-rr 29.0%

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

   4. Taylor expanded in c0 around -inf 0.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + {c0}^{2} \cdot \left(-0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} + 0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. unpow2 0.0%

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

      4. distribute-rgt-out 0.0%

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

      5. metadata-eval 0.0%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot \color{blue}{0}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      6. mul0-rgt 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, \color{blue}{0}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      7. associate-*r/ 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}}\right) \]

      8. unpow2 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}\right) \]

      9. *-commutative 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}}\right) \]

      10. unpow2 59.2%

          \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}}\right) \]

      11. unpow2 59.2%

          \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}}\right) \]

   6. Simplified 59.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -9.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{elif}\;w \leq -6.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ \mathbf{elif}\;w \leq -7 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}\right)\\ \mathbf{elif}\;w \leq 3.25 \cdot 10^{+25}:\\ \;\;\;\;\frac{c0}{w \cdot h} \cdot \left(\frac{c0}{2 \cdot w} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \end{array} \]

Alternative 4: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)}{2 \cdot w}\\ t_1 := \frac{0}{w \cdot h}\\ \mathbf{if}\;w \leq -2.5 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \leq -6.4 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -8.5 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}\right)\\ \mathbf{elif}\;w \leq 8.2 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* (/ c0 (* w h)) (* 2.0 (pow (/ d D) 2.0)))) (* 2.0 w)))
        (t_1 (/ 0.0 (* w h))))
   (if (<= w -2.5e+68)
     t_1
     (if (<= w -6.4e-114)
       t_0
       (if (<= w -8.5e-177)
         (fma (* c0 c0) 0.0 (/ (* 0.25 (* (* D D) (* h (* M M)))) (* d d)))
         (if (<= w 8.2e+20) t_0 t_1))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * ((c0 / (w * h)) * (2.0 * pow((d / D), 2.0)))) / (2.0 * w);
	double t_1 = 0.0 / (w * h);
	double tmp;
	if (w <= -2.5e+68) {
		tmp = t_1;
	} else if (w <= -6.4e-114) {
		tmp = t_0;
	} else if (w <= -8.5e-177) {
		tmp = fma((c0 * c0), 0.0, ((0.25 * ((D * D) * (h * (M * M)))) / (d * d)));
	} else if (w <= 8.2e+20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(Float64(c0 / Float64(w * h)) * Float64(2.0 * (Float64(d / D) ^ 2.0)))) / Float64(2.0 * w))
	t_1 = Float64(0.0 / Float64(w * h))
	tmp = 0.0
	if (w <= -2.5e+68)
		tmp = t_1;
	elseif (w <= -6.4e-114)
		tmp = t_0;
	elseif (w <= -8.5e-177)
		tmp = fma(Float64(c0 * c0), 0.0, Float64(Float64(0.25 * Float64(Float64(D * D) * Float64(h * Float64(M * M)))) / Float64(d * d)));
	elseif (w <= 8.2e+20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Power[N[(d / D), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -2.5e+68], t$95$1, If[LessEqual[w, -6.4e-114], t$95$0, If[LessEqual[w, -8.5e-177], N[(N[(c0 * c0), $MachinePrecision] * 0.0 + N[(N[(0.25 * N[(N[(D * D), $MachinePrecision] * N[(h * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[w, 8.2e+20], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if w < -2.5000000000000002e68 or 8.2e20 < w

   1. Initial program 18.0%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in c0 around -inf 4.7%

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

      1. associate-*r/ 4.7%

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

      2. unpow2 4.7%

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

      3. neg-mul-1 4.7%

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

      4. associate-*r* 5.7%

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

      5. unpow2 5.7%

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

   6. Simplified 5.7%

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

   7. Taylor expanded in h around 0 10.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 10.3%

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

   9. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]

   if -2.5000000000000002e68 < w < -6.4000000000000003e-114 or -8.4999999999999993e-177 < w < 8.2e20

   1. Initial program 31.5%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in D around 0 9.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{0.5 \cdot \frac{h \cdot \left(w \cdot \left(-1 \cdot \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w} + \frac{M \cdot \left(c0 \cdot {d}^{2}\right)}{h \cdot w}\right)\right)}{c0 \cdot {d}^{2}} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}\right) \]
   5. Step-by-step derivation

      1. fma-def 9.0%

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

   6. Simplified 10.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{\frac{c0}{h}}{w}, \frac{d}{D} \cdot \frac{d}{D}, \color{blue}{\mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)}\right) \]
   7. Step-by-step derivation

      1. fma-udef 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right)} \]

      2. associate-/r* 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\frac{c0}{h \cdot w}} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      3. pow2 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\frac{c0}{h \cdot w} \cdot \color{blue}{{\left(\frac{d}{D}\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      4. *-commutative 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{{\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{h \cdot w}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

      5. *-commutative 10.4%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{\color{blue}{w \cdot h}} + \mathsf{fma}\left(0.5, \frac{\left(h \cdot w\right) \cdot \left(0 \cdot \left(\frac{M}{h} \cdot \frac{\left(c0 \cdot d\right) \cdot d}{w}\right)\right)}{\left(c0 \cdot d\right) \cdot d}, \frac{\frac{c0}{h \cdot w}}{\frac{D \cdot D}{d \cdot d}}\right)\right) \]

   8. Applied egg-rr 44.9%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + \mathsf{fma}\left(0.5, \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)}, {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right) + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)} \]

      2. fma-udef 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right) \]

      3. associate-+l+ 44.9%

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{h \cdot \left(w \cdot 0\right)}{c0 \cdot \left(d \cdot d\right)} + \left({\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h} + {\left(\frac{d}{D}\right)}^{2} \cdot \frac{c0}{w \cdot h}\right)\right)} \]

   10. Simplified 46.5%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\frac{h}{c0} \cdot 0 + \left(\frac{c0}{h \cdot w} \cdot {\left(\frac{d}{D}\right)}^{2}\right) \cdot 2\right)} \]
   11. Step-by-step derivation

       1. associate-*l/ 46.5%

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

       2. mul0-rgt 48.3%

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

       3. pow2 48.3%

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

       4. associate-*l* 48.3%

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

       5. pow2 48.3%

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

       6. *-commutative 48.3%

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

   12. Applied egg-rr 48.3%

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

   if -6.4000000000000003e-114 < w < -8.4999999999999993e-177

   1. Initial program 21.8%

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

      1. distribute-lft-in 21.8%

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

      2. *-commutative 21.8%

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

      3. times-frac 21.5%

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

      4. frac-times 22.0%

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

      5. pow2 22.0%

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

      6. *-commutative 22.0%

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

   3. Applied egg-rr 29.0%

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

   4. Taylor expanded in c0 around -inf 0.0%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + {c0}^{2} \cdot \left(-0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} + 0.5 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 0.0%

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

      2. fma-def 0.0%

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

      3. unpow2 0.0%

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

      4. distribute-rgt-out 0.0%

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

      5. metadata-eval 0.0%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot {w}^{2}\right)} \cdot \color{blue}{0}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      6. mul0-rgt 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, \color{blue}{0}, 0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}\right) \]

      7. associate-*r/ 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \color{blue}{\frac{0.25 \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}}\right) \]

      8. unpow2 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}\right) \]

      9. *-commutative 59.2%

         \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right)}{{d}^{2}}\right) \]

      10. unpow2 59.2%

          \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right)}{{d}^{2}}\right) \]

      11. unpow2 59.2%

          \[\leadsto \mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{\color{blue}{d \cdot d}}\right) \]

   6. Simplified 59.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{elif}\;w \leq -6.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)}{2 \cdot w}\\ \mathbf{elif}\;w \leq -8.5 \cdot 10^{-177}:\\ \;\;\;\;\mathsf{fma}\left(c0 \cdot c0, 0, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot d}\right)\\ \mathbf{elif}\;w \leq 8.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{c0}{w \cdot h} \cdot \left(2 \cdot {\left(\frac{d}{D}\right)}^{2}\right)\right)}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \end{array} \]

Alternative 5: 37.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.6 \cdot 10^{+67} \lor \neg \left(w \leq -7.6 \cdot 10^{-111}\right) \land \left(w \leq -1.12 \cdot 10^{-166} \lor \neg \left(w \leq 8.2 \cdot 10^{-38}\right)\right):\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{h \cdot \left(D \cdot D\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= w -4.6e+67)
         (and (not (<= w -7.6e-111))
              (or (<= w -1.12e-166) (not (<= w 8.2e-38)))))
   (/ 0.0 (* w h))
   (* (/ c0 (* 2.0 w)) (* 2.0 (* (/ c0 w) (/ (* d d) (* h (* D D))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -4.6e+67) || (!(w <= -7.6e-111) && ((w <= -1.12e-166) || !(w <= 8.2e-38)))) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((w <= (-4.6d+67)) .or. (.not. (w <= (-7.6d-111))) .and. (w <= (-1.12d-166)) .or. (.not. (w <= 8.2d-38))) then
        tmp = 0.0d0 / (w * h)
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * ((c0 / w) * ((d_1 * d_1) / (h * (d * d)))))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((w <= -4.6e+67) || (!(w <= -7.6e-111) && ((w <= -1.12e-166) || !(w <= 8.2e-38)))) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (w <= -4.6e+67) or (not (w <= -7.6e-111) and ((w <= -1.12e-166) or not (w <= 8.2e-38))):
		tmp = 0.0 / (w * h)
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((w <= -4.6e+67) || (!(w <= -7.6e-111) && ((w <= -1.12e-166) || !(w <= 8.2e-38))))
		tmp = Float64(0.0 / Float64(w * h));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(Float64(c0 / w) * Float64(Float64(d * d) / Float64(h * Float64(D * D))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((w <= -4.6e+67) || (~((w <= -7.6e-111)) && ((w <= -1.12e-166) || ~((w <= 8.2e-38)))))
		tmp = 0.0 / (w * h);
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * ((c0 / w) * ((d * d) / (h * (D * D)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[w, -4.6e+67], And[N[Not[LessEqual[w, -7.6e-111]], $MachinePrecision], Or[LessEqual[w, -1.12e-166], N[Not[LessEqual[w, 8.2e-38]], $MachinePrecision]]]], N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(c0 / w), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4.5999999999999997e67 or -7.60000000000000045e-111 < w < -1.11999999999999994e-166 or 8.1999999999999996e-38 < w

   1. Initial program 18.1%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in c0 around -inf 4.7%

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

      1. associate-*r/ 4.7%

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

      2. unpow2 4.7%

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

      3. neg-mul-1 4.7%

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

      4. associate-*r* 5.5%

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

      5. unpow2 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in h around 0 9.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 9.2%

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

   9. Simplified 48.4%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]

   if -4.5999999999999997e67 < w < -7.60000000000000045e-111 or -1.11999999999999994e-166 < w < 8.1999999999999996e-38

   1. Initial program 32.5%

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

   3. Simplified 33.0%

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

   4. Taylor expanded in c0 around inf 44.3%

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

      1. associate-*r/ 44.3%

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

      2. associate-*r* 42.8%

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

      3. *-commutative 42.8%

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

      4. unpow2 42.8%

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

      5. *-commutative 42.8%

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

      6. associate-*r/ 42.8%

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

      7. times-frac 44.7%

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

      8. unpow2 44.7%

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

      9. unpow2 44.7%

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

      10. *-commutative 44.7%

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

      11. unpow2 44.7%

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

   6. Simplified 44.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -4.6 \cdot 10^{+67} \lor \neg \left(w \leq -7.6 \cdot 10^{-111}\right) \land \left(w \leq -1.12 \cdot 10^{-166} \lor \neg \left(w \leq 8.2 \cdot 10^{-38}\right)\right):\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(\frac{c0}{w} \cdot \frac{d \cdot d}{h \cdot \left(D \cdot D\right)}\right)\right)\\ \end{array} \]

Alternative 6: 33.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{if}\;c0 \leq -8.6 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 3.9 \cdot 10^{+110}:\\ \;\;\;\;0 \cdot \left(c0 \cdot c0\right)\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{+194} \lor \neg \left(c0 \leq 1.2 \cdot 10^{+274}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* c0 c0) (* D D)) (/ (* d d) (* h (* w w))))))
   (if (<= c0 -8.6e-126)
     t_0
     (if (<= c0 3.9e+110)
       (* 0.0 (* c0 c0))
       (if (or (<= c0 1.15e+194) (not (<= c0 1.2e+274)))
         t_0
         (/ 0.0 (* w h)))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	double tmp;
	if (c0 <= -8.6e-126) {
		tmp = t_0;
	} else if (c0 <= 3.9e+110) {
		tmp = 0.0 * (c0 * c0);
	} else if ((c0 <= 1.15e+194) || !(c0 <= 1.2e+274)) {
		tmp = t_0;
	} else {
		tmp = 0.0 / (w * h);
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c0 * c0) / (d * d)) * ((d_1 * d_1) / (h * (w * w)))
    if (c0 <= (-8.6d-126)) then
        tmp = t_0
    else if (c0 <= 3.9d+110) then
        tmp = 0.0d0 * (c0 * c0)
    else if ((c0 <= 1.15d+194) .or. (.not. (c0 <= 1.2d+274))) then
        tmp = t_0
    else
        tmp = 0.0d0 / (w * h)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	double tmp;
	if (c0 <= -8.6e-126) {
		tmp = t_0;
	} else if (c0 <= 3.9e+110) {
		tmp = 0.0 * (c0 * c0);
	} else if ((c0 <= 1.15e+194) || !(c0 <= 1.2e+274)) {
		tmp = t_0;
	} else {
		tmp = 0.0 / (w * h);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)))
	tmp = 0
	if c0 <= -8.6e-126:
		tmp = t_0
	elif c0 <= 3.9e+110:
		tmp = 0.0 * (c0 * c0)
	elif (c0 <= 1.15e+194) or not (c0 <= 1.2e+274):
		tmp = t_0
	else:
		tmp = 0.0 / (w * h)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(c0 * c0) / Float64(D * D)) * Float64(Float64(d * d) / Float64(h * Float64(w * w))))
	tmp = 0.0
	if (c0 <= -8.6e-126)
		tmp = t_0;
	elseif (c0 <= 3.9e+110)
		tmp = Float64(0.0 * Float64(c0 * c0));
	elseif ((c0 <= 1.15e+194) || !(c0 <= 1.2e+274))
		tmp = t_0;
	else
		tmp = Float64(0.0 / Float64(w * h));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	tmp = 0.0;
	if (c0 <= -8.6e-126)
		tmp = t_0;
	elseif (c0 <= 3.9e+110)
		tmp = 0.0 * (c0 * c0);
	elseif ((c0 <= 1.15e+194) || ~((c0 <= 1.2e+274)))
		tmp = t_0;
	else
		tmp = 0.0 / (w * h);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(c0 * c0), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -8.6e-126], t$95$0, If[LessEqual[c0, 3.9e+110], N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c0, 1.15e+194], N[Not[LessEqual[c0, 1.2e+274]], $MachinePrecision]], t$95$0, N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c0 < -8.60000000000000065e-126 or 3.9000000000000003e110 < c0 < 1.15000000000000003e194 or 1.2e274 < c0

   1. Initial program 35.4%

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

      1. distribute-lft-in 35.3%

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

      2. *-commutative 35.3%

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

      3. times-frac 33.8%

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

      4. frac-times 33.0%

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

      5. pow2 33.0%

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

      6. *-commutative 33.0%

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

   3. Applied egg-rr 37.3%

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

   4. Taylor expanded in c0 around inf 43.8%

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

      1. times-frac 43.0%

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

      2. unpow2 43.0%

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

      3. unpow2 43.0%

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

      4. unpow2 43.0%

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

      5. unpow2 43.0%

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

   6. Simplified 43.0%

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

   if -8.60000000000000065e-126 < c0 < 3.9000000000000003e110

   1. Initial program 18.3%

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

      1. distribute-lft-in 18.3%

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

      2. *-commutative 18.3%

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

      3. times-frac 15.6%

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

      4. frac-times 16.7%

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

      5. pow2 16.7%

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

      6. *-commutative 16.7%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in c0 around -inf 7.6%

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

      1. unpow2 7.6%

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

      2. distribute-rgt-out 7.6%

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

      3. metadata-eval 7.6%

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

      4. mul0-rgt 43.5%

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

   6. Simplified 43.5%

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

   if 1.15000000000000003e194 < c0 < 1.2e274

   1. Initial program 10.0%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in c0 around -inf 0.1%

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

      1. associate-*r/ 0.1%

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

      2. unpow2 0.1%

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

      3. neg-mul-1 0.1%

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

      4. associate-*r* 0.1%

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

      5. unpow2 0.1%

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

   6. Simplified 0.1%

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

   7. Taylor expanded in h around 0 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

   9. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -8.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 3.9 \cdot 10^{+110}:\\ \;\;\;\;0 \cdot \left(c0 \cdot c0\right)\\ \mathbf{elif}\;c0 \leq 1.15 \cdot 10^{+194} \lor \neg \left(c0 \leq 1.2 \cdot 10^{+274}\right):\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \end{array} \]

Alternative 7: 33.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{if}\;c0 \leq -2.55 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c0 \leq 2 \cdot 10^{+107}:\\ \;\;\;\;0 \cdot \left(c0 \cdot c0\right)\\ \mathbf{elif}\;c0 \leq 5.4 \cdot 10^{+200}:\\ \;\;\;\;\frac{c0 \cdot c0}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(w \cdot w\right)}{d \cdot d}}\\ \mathbf{elif}\;c0 \leq 10^{+274}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* c0 c0) (* D D)) (/ (* d d) (* h (* w w))))))
   (if (<= c0 -2.55e-121)
     t_0
     (if (<= c0 2e+107)
       (* 0.0 (* c0 c0))
       (if (<= c0 5.4e+200)
         (/ (* c0 c0) (/ (* (* h (* D D)) (* w w)) (* d d)))
         (if (<= c0 1e+274) (/ 0.0 (* w h)) t_0))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	double tmp;
	if (c0 <= -2.55e-121) {
		tmp = t_0;
	} else if (c0 <= 2e+107) {
		tmp = 0.0 * (c0 * c0);
	} else if (c0 <= 5.4e+200) {
		tmp = (c0 * c0) / (((h * (D * D)) * (w * w)) / (d * d));
	} else if (c0 <= 1e+274) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c0 * c0) / (d * d)) * ((d_1 * d_1) / (h * (w * w)))
    if (c0 <= (-2.55d-121)) then
        tmp = t_0
    else if (c0 <= 2d+107) then
        tmp = 0.0d0 * (c0 * c0)
    else if (c0 <= 5.4d+200) then
        tmp = (c0 * c0) / (((h * (d * d)) * (w * w)) / (d_1 * d_1))
    else if (c0 <= 1d+274) then
        tmp = 0.0d0 / (w * h)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	double tmp;
	if (c0 <= -2.55e-121) {
		tmp = t_0;
	} else if (c0 <= 2e+107) {
		tmp = 0.0 * (c0 * c0);
	} else if (c0 <= 5.4e+200) {
		tmp = (c0 * c0) / (((h * (D * D)) * (w * w)) / (d * d));
	} else if (c0 <= 1e+274) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)))
	tmp = 0
	if c0 <= -2.55e-121:
		tmp = t_0
	elif c0 <= 2e+107:
		tmp = 0.0 * (c0 * c0)
	elif c0 <= 5.4e+200:
		tmp = (c0 * c0) / (((h * (D * D)) * (w * w)) / (d * d))
	elif c0 <= 1e+274:
		tmp = 0.0 / (w * h)
	else:
		tmp = t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(c0 * c0) / Float64(D * D)) * Float64(Float64(d * d) / Float64(h * Float64(w * w))))
	tmp = 0.0
	if (c0 <= -2.55e-121)
		tmp = t_0;
	elseif (c0 <= 2e+107)
		tmp = Float64(0.0 * Float64(c0 * c0));
	elseif (c0 <= 5.4e+200)
		tmp = Float64(Float64(c0 * c0) / Float64(Float64(Float64(h * Float64(D * D)) * Float64(w * w)) / Float64(d * d)));
	elseif (c0 <= 1e+274)
		tmp = Float64(0.0 / Float64(w * h));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = ((c0 * c0) / (D * D)) * ((d * d) / (h * (w * w)));
	tmp = 0.0;
	if (c0 <= -2.55e-121)
		tmp = t_0;
	elseif (c0 <= 2e+107)
		tmp = 0.0 * (c0 * c0);
	elseif (c0 <= 5.4e+200)
		tmp = (c0 * c0) / (((h * (D * D)) * (w * w)) / (d * d));
	elseif (c0 <= 1e+274)
		tmp = 0.0 / (w * h);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(c0 * c0), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c0, -2.55e-121], t$95$0, If[LessEqual[c0, 2e+107], N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 5.4e+200], N[(N[(c0 * c0), $MachinePrecision] / N[(N[(N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c0, 1e+274], N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c0 < -2.5499999999999999e-121 or 9.99999999999999921e273 < c0

   1. Initial program 35.4%

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

      1. distribute-lft-in 35.3%

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

      2. *-commutative 35.3%

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

      3. times-frac 34.4%

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

      4. frac-times 33.5%

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

      5. pow2 33.5%

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

      6. *-commutative 33.5%

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

   3. Applied egg-rr 37.6%

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

   4. Taylor expanded in c0 around inf 43.1%

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

      1. times-frac 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. unpow2 42.2%

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

      5. unpow2 42.2%

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

   6. Simplified 42.2%

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

   if -2.5499999999999999e-121 < c0 < 1.9999999999999999e107

   1. Initial program 18.3%

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

      1. distribute-lft-in 18.3%

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

      2. *-commutative 18.3%

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

      3. times-frac 15.6%

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

      4. frac-times 16.7%

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

      5. pow2 16.7%

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

      6. *-commutative 16.7%

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

   3. Applied egg-rr 28.3%

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

   4. Taylor expanded in c0 around -inf 7.6%

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

      1. unpow2 7.6%

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

      2. distribute-rgt-out 7.6%

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

      3. metadata-eval 7.6%

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

      4. mul0-rgt 43.5%

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

   6. Simplified 43.5%

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

   if 1.9999999999999999e107 < c0 < 5.40000000000000031e200

   1. Initial program 33.5%

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

   2. Taylor expanded in c0 around inf 45.3%

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

      1. associate-/l* 45.3%

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

      2. unpow2 45.3%

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

      3. associate-*r* 50.7%

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

      4. unpow2 50.7%

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

      5. unpow2 50.7%

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

      6. unpow2 50.7%

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

   4. Simplified 50.7%

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

   if 5.40000000000000031e200 < c0 < 9.99999999999999921e273

   1. Initial program 10.5%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in c0 around -inf 0.1%

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

      1. associate-*r/ 0.1%

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

      2. unpow2 0.1%

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

      3. neg-mul-1 0.1%

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

      4. associate-*r* 0.1%

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

      5. unpow2 0.1%

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

   6. Simplified 0.1%

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

   7. Taylor expanded in h around 0 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

   9. Simplified 34.3%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq -2.55 \cdot 10^{-121}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \mathbf{elif}\;c0 \leq 2 \cdot 10^{+107}:\\ \;\;\;\;0 \cdot \left(c0 \cdot c0\right)\\ \mathbf{elif}\;c0 \leq 5.4 \cdot 10^{+200}:\\ \;\;\;\;\frac{c0 \cdot c0}{\frac{\left(h \cdot \left(D \cdot D\right)\right) \cdot \left(w \cdot w\right)}{d \cdot d}}\\ \mathbf{elif}\;c0 \leq 10^{+274}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot c0}{D \cdot D} \cdot \frac{d \cdot d}{h \cdot \left(w \cdot w\right)}\\ \end{array} \]

Alternative 8: 19.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \cdot d \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{c0}{\frac{w}{\left(d \cdot d\right) \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* d d) INFINITY)
   (* 0.5 (/ c0 (/ w (* (* d d) 0.0))))
   (* (/ c0 (* 2.0 w)) 0.0)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= ((double) INFINITY)) {
		tmp = 0.5 * (c0 / (w / ((d * d) * 0.0)));
	} else {
		tmp = (c0 / (2.0 * w)) * 0.0;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d * d) <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (c0 / (w / ((d * d) * 0.0)));
	} else {
		tmp = (c0 / (2.0 * w)) * 0.0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d * d) <= math.inf:
		tmp = 0.5 * (c0 / (w / ((d * d) * 0.0)))
	else:
		tmp = (c0 / (2.0 * w)) * 0.0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(d * d) <= Inf)
		tmp = Float64(0.5 * Float64(c0 / Float64(w / Float64(Float64(d * d) * 0.0))));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d * d) <= Inf)
		tmp = 0.5 * (c0 / (w / ((d * d) * 0.0)));
	else
		tmp = (c0 / (2.0 * w)) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(d * d), $MachinePrecision], Infinity], N[(0.5 * N[(c0 / N[(w / N[(N[(d * d), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 d d) < +inf.0

   1. Initial program 26.2%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in c0 around -inf 3.5%

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

      1. associate-*r/ 3.5%

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

      2. unpow2 3.5%

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

      3. neg-mul-1 3.5%

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

      4. associate-*r* 4.3%

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

      5. unpow2 4.3%

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

   6. Simplified 4.3%

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

      1. frac-times 4.3%

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

   8. Applied egg-rr 4.3%

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

   9. Taylor expanded in d around 0 5.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(-1 \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)\right)}{w}} \]
   10. Step-by-step derivation

       1. associate-/l* 5.0%

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

       2. unpow2 5.0%

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

       3. distribute-lft1-in 5.0%

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

       4. metadata-eval 5.0%

          \[\leadsto 0.5 \cdot \frac{c0}{\frac{w}{\left(d \cdot d\right) \cdot \left(\color{blue}{0} \cdot \frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}\right)}} \]

       5. mul0-lft 18.1%

          \[\leadsto 0.5 \cdot \frac{c0}{\frac{w}{\left(d \cdot d\right) \cdot \color{blue}{0}}} \]

   11. Simplified 18.1%

       \[\leadsto \color{blue}{0.5 \cdot \frac{c0}{\frac{w}{\left(d \cdot d\right) \cdot 0}}} \]

   if +inf.0 < (*.f64 d d)

   1. Initial program 26.2%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in c0 around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

      2. distribute-lft-in 3.8%

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

   6. Simplified 29.3%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \cdot d \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{c0}{\frac{w}{\left(d \cdot d\right) \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot 0\\ \end{array} \]

Alternative 9: 30.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \cdot D \leq 3 \cdot 10^{-94}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot \left(D \cdot D\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* D D) 3e-94) (/ 0.0 (* w h)) (/ 0.0 (* w (* D D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 3e-94) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = 0.0 / (w * (D * D));
	}
	return tmp;
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((d * d) <= 3d-94) then
        tmp = 0.0d0 / (w * h)
    else
        tmp = 0.0d0 / (w * (d * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((D * D) <= 3e-94) {
		tmp = 0.0 / (w * h);
	} else {
		tmp = 0.0 / (w * (D * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (D * D) <= 3e-94:
		tmp = 0.0 / (w * h)
	else:
		tmp = 0.0 / (w * (D * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(D * D) <= 3e-94)
		tmp = Float64(0.0 / Float64(w * h));
	else
		tmp = Float64(0.0 / Float64(w * Float64(D * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((D * D) <= 3e-94)
		tmp = 0.0 / (w * h);
	else
		tmp = 0.0 / (w * (D * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(D * D), $MachinePrecision], 3e-94], N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(w * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 D D) < 3.0000000000000001e-94

   1. Initial program 25.3%

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

   3. Simplified 25.1%

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

   4. Taylor expanded in c0 around -inf 2.0%

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

      1. associate-*r/ 2.0%

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

      2. unpow2 2.0%

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

      3. neg-mul-1 2.0%

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

      4. associate-*r* 2.6%

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

      5. unpow2 2.6%

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

   6. Simplified 2.6%

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

   7. Taylor expanded in h around 0 5.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 5.1%

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

   9. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]

   if 3.0000000000000001e-94 < (*.f64 D D)

   1. Initial program 27.9%

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

   3. Simplified 27.9%

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

   4. Taylor expanded in c0 around -inf 6.0%

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

      1. associate-*r/ 6.0%

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

      2. unpow2 6.0%

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

      3. neg-mul-1 6.0%

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

      4. associate-*r* 7.5%

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

      5. unpow2 7.5%

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

   6. Simplified 7.5%

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

   7. Taylor expanded in D around 0 4.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{h \cdot w} + \frac{c0 \cdot {d}^{2}}{h \cdot w}\right)}{{D}^{2} \cdot w}} \]
   8. Step-by-step derivation

      1. associate-*r/ 4.7%

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

   9. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{0}{\left(D \cdot D\right) \cdot w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \cdot D \leq 3 \cdot 10^{-94}:\\ \;\;\;\;\frac{0}{w \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w \cdot \left(D \cdot D\right)}\\ \end{array} \]

Alternative 10: 25.6% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ 0 \cdot \left(c0 \cdot c0\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* 0.0 (* c0 c0)))

C

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

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0 * (c0 * c0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0 * (c0 * c0);
}

Python

def code(c0, w, h, D, d, M):
	return 0.0 * (c0 * c0)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.0 * Float64(c0 * c0))
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.0 * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.2%

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

   1. distribute-lft-in 26.2%

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

   2. *-commutative 26.2%

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

   3. times-frac 24.3%

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

   4. frac-times 24.3%

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

   5. pow2 24.3%

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

   6. *-commutative 24.3%

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

3. Applied egg-rr 31.4%

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

4. Taylor expanded in c0 around -inf 4.4%

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

   1. unpow2 4.4%

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

   2. distribute-rgt-out 4.4%

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

   3. metadata-eval 4.4%

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

   4. mul0-rgt 25.1%

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

6. Simplified 25.1%

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

7. Final simplification 25.1%

   \[\leadsto 0 \cdot \left(c0 \cdot c0\right) \]

Alternative 11: 28.8% accurate, 30.2× speedup

Math

\[\begin{array}{l} \\ \frac{0}{w \cdot h} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (/ 0.0 (* w h)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0 / (w * h);
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0 / (w * h)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0 / (w * h);
}

Python

def code(c0, w, h, D, d, M):
	return 0.0 / (w * h)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.0 / Float64(w * h))
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.0 / N[(w * h), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.2%

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

3. Simplified 26.1%

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

4. Taylor expanded in c0 around -inf 3.5%

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

   1. associate-*r/ 3.5%

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

   2. unpow2 3.5%

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

   3. neg-mul-1 3.5%

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

   4. associate-*r* 4.3%

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

   5. unpow2 4.3%

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

6. Simplified 4.3%

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

7. Taylor expanded in h around 0 5.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(-1 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w} + \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot w}\right)}{h \cdot w}} \]
8. Step-by-step derivation

   1. associate-*r/ 5.0%

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

9. Simplified 30.0%

   \[\leadsto \color{blue}{\frac{0}{h \cdot w}} \]

10. Final simplification 30.0%

    \[\leadsto \frac{0}{w \cdot h} \]

Reproduce

herbie shell --seed 2023277 
(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))))))