Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 73.3%

Time: 27.0s

Alternatives: 4

Speedup: 21.6×

• could not determine a ground truth

  1. c0 = 5.044694549083937e+223
  2. w = 4.062885313806089e-284
  3. h = -2.40143405275751e-173
  4. D = -1.2393883466329227e-307
  5. d = -1.3006695509753214e+237
  6. M = -1.0484087802910232e+87

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: 24.5% 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: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ t_3 := \frac{D \cdot M}{d}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-44}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{w \cdot h}\right)\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq \infty\right):\\ \;\;\;\;0.25 \cdot \left(h \cdot \left(t\_3 \cdot t\_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))))
        (t_3 (/ (* D M) d)))
   (if (<= t_2 -1e-44)
     (* t_0 (* 2.0 (* c0 (* (/ d D) (/ (/ d D) (* w h))))))
     (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
       (* 0.25 (* h (* t_3 t_3)))
       (* t_0 (* 2.0 (* c0 (* (/ (/ d D) w) (/ (/ d D) h)))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double t_3 = (D * M) / d;
	double tmp;
	if (t_2 <= -1e-44) {
		tmp = t_0 * (2.0 * (c0 * ((d / D) * ((d / D) / (w * h)))));
	} else if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = 0.25 * (h * (t_3 * t_3));
	} else {
		tmp = t_0 * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double t_3 = (D * M) / d;
	double tmp;
	if (t_2 <= -1e-44) {
		tmp = t_0 * (2.0 * (c0 * ((d / D) * ((d / D) / (w * h)))));
	} else if ((t_2 <= 0.0) || !(t_2 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.25 * (h * (t_3 * t_3));
	} else {
		tmp = t_0 * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	t_3 = (D * M) / d
	tmp = 0
	if t_2 <= -1e-44:
		tmp = t_0 * (2.0 * (c0 * ((d / D) * ((d / D) / (w * h)))))
	elif (t_2 <= 0.0) or not (t_2 <= math.inf):
		tmp = 0.25 * (h * (t_3 * t_3))
	else:
		tmp = t_0 * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	t_3 = Float64(Float64(D * M) / d)
	tmp = 0.0
	if (t_2 <= -1e-44)
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 * Float64(Float64(d / D) * Float64(Float64(d / D) / Float64(w * h))))));
	elseif ((t_2 <= 0.0) || !(t_2 <= Inf))
		tmp = Float64(0.25 * Float64(h * Float64(t_3 * t_3)));
	else
		tmp = Float64(t_0 * Float64(2.0 * Float64(c0 * Float64(Float64(Float64(d / D) / w) * Float64(Float64(d / D) / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	t_3 = (D * M) / d;
	tmp = 0.0;
	if (t_2 <= -1e-44)
		tmp = t_0 * (2.0 * (c0 * ((d / D) * ((d / D) / (w * h)))));
	elseif ((t_2 <= 0.0) || ~((t_2 <= Inf)))
		tmp = 0.25 * (h * (t_3 * t_3));
	else
		tmp = t_0 * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-44], N[(t$95$0 * N[(2.0 * N[(c0 * N[(N[(d / D), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, Infinity]], $MachinePrecision]], N[(0.25 * N[(h * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[(c0 * N[(N[(N[(d / D), $MachinePrecision] / w), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -9.99999999999999953e-45

   1. Initial program 81.2%

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

   3. Simplified 81.2%

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

      1. times-frac 81.2%

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in c0 around inf 84.0%

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

      1. associate-/l* 84.1%

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

      2. associate-/r* 84.1%

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

      3. unpow2 84.1%

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

      4. unpow2 84.1%

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

      5. times-frac 84.1%

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

      6. unpow2 84.1%

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

   9. Simplified 84.1%

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

       1. unpow2 84.1%

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

       2. *-un-lft-identity 84.1%

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

       3. times-frac 89.4%

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

   11. Applied egg-rr 89.4%

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

   if -9.99999999999999953e-45 < (*.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))))) < -0.0 or +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 3.0%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in c0 around -inf 7.6%

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

      1. +-commutative 7.6%

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

      2. associate-/l* 8.0%

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

      3. *-commutative 8.0%

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

      4. distribute-lft-in 6.2%

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

   7. Simplified 34.7%

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

   8. Taylor expanded in c0 around 0 51.4%

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

      1. associate-/l* 51.3%

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

      2. pow2 51.3%

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

      3. *-commutative 51.3%

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

      4. pow2 51.3%

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

   10. Applied egg-rr 51.3%

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

       1. associate-*r/ 51.4%

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

       2. *-commutative 51.4%

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

       3. associate-*r* 51.7%

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

       4. unpow2 51.7%

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

       5. unpow2 51.7%

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

       6. swap-sqr 61.2%

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

       7. unpow2 61.2%

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

       8. *-commutative 61.2%

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

       9. associate-/l* 61.2%

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

       10. *-commutative 61.2%

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

   12. Simplified 61.2%

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

       1. unpow2 61.2%

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

       2. unpow2 61.2%

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

       3. times-frac 74.2%

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

   14. Applied egg-rr 74.2%

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

   if -0.0 < (*.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))))) < +inf.0

   1. Initial program 84.9%

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

   3. Simplified 80.6%

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

      1. times-frac 80.5%

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

   6. Applied egg-rr 80.5%

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

   7. Taylor expanded in c0 around inf 83.7%

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

      1. associate-/l* 81.5%

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

      2. associate-/r* 79.3%

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

      3. unpow2 79.3%

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

      4. unpow2 79.3%

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

      5. times-frac 81.4%

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

      6. unpow2 81.4%

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

   9. Simplified 81.4%

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

       1. unpow2 81.4%

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

       2. *-commutative 81.4%

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

       3. times-frac 85.6%

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

   11. Applied egg-rr 85.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1 \cdot 10^{-44}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{d}{D} \cdot \frac{\frac{d}{D}}{w \cdot h}\right)\right)\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;0.25 \cdot \left(h \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d}\\ \mathbf{if}\;d \leq 9 \cdot 10^{+223} \lor \neg \left(d \leq 4.4 \cdot 10^{+295}\right):\\ \;\;\;\;0.25 \cdot \left(h \cdot \left(t\_0 \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* D M) d)))
   (if (or (<= d 9e+223) (not (<= d 4.4e+295)))
     (* 0.25 (* h (* t_0 t_0)))
     (* (/ c0 (* 2.0 w)) (* 2.0 (* c0 (* (/ (/ d D) w) (/ (/ d D) h))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * M) / d;
	double tmp;
	if ((d <= 9e+223) || !(d <= 4.4e+295)) {
		tmp = 0.25 * (h * (t_0 * t_0));
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d / D) / w) * ((d / D) / 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 = (d * m) / d_1
    if ((d_1 <= 9d+223) .or. (.not. (d_1 <= 4.4d+295))) then
        tmp = 0.25d0 * (h * (t_0 * t_0))
    else
        tmp = (c0 / (2.0d0 * w)) * (2.0d0 * (c0 * (((d_1 / d) / w) * ((d_1 / d) / 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 = (D * M) / d;
	double tmp;
	if ((d <= 9e+223) || !(d <= 4.4e+295)) {
		tmp = 0.25 * (h * (t_0 * t_0));
	} else {
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (D * M) / d
	tmp = 0
	if (d <= 9e+223) or not (d <= 4.4e+295):
		tmp = 0.25 * (h * (t_0 * t_0))
	else:
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * M) / d)
	tmp = 0.0
	if ((d <= 9e+223) || !(d <= 4.4e+295))
		tmp = Float64(0.25 * Float64(h * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(2.0 * Float64(c0 * Float64(Float64(Float64(d / D) / w) * Float64(Float64(d / D) / h)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * M) / d;
	tmp = 0.0;
	if ((d <= 9e+223) || ~((d <= 4.4e+295)))
		tmp = 0.25 * (h * (t_0 * t_0));
	else
		tmp = (c0 / (2.0 * w)) * (2.0 * (c0 * (((d / D) / w) * ((d / D) / h))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]}, If[Or[LessEqual[d, 9e+223], N[Not[LessEqual[d, 4.4e+295]], $MachinePrecision]], N[(0.25 * N[(h * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 * N[(N[(N[(d / D), $MachinePrecision] / w), $MachinePrecision] * N[(N[(d / D), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 9e223 or 4.3999999999999996e295 < d

   1. Initial program 27.3%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in c0 around -inf 7.3%

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

      1. +-commutative 7.3%

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

      2. associate-/l* 7.6%

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

      3. *-commutative 7.6%

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

      4. distribute-lft-in 6.2%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in c0 around 0 41.5%

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

      1. associate-/l* 41.4%

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

      2. pow2 41.4%

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

      3. *-commutative 41.4%

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

      4. pow2 41.4%

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

   10. Applied egg-rr 41.4%

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

       1. associate-*r/ 41.5%

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

       2. *-commutative 41.5%

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

       3. associate-*r* 42.7%

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

       4. unpow2 42.7%

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

       5. unpow2 42.7%

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

       6. swap-sqr 49.1%

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

       7. unpow2 49.1%

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

       8. *-commutative 49.1%

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

       9. associate-/l* 49.1%

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

       10. *-commutative 49.1%

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

   12. Simplified 49.1%

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

       1. unpow2 49.1%

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

       2. unpow2 49.1%

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

       3. times-frac 56.7%

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

   14. Applied egg-rr 56.7%

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

   if 9e223 < d < 4.3999999999999996e295

   1. Initial program 38.2%

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

   3. Simplified 38.2%

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

      1. times-frac 38.2%

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

   6. Applied egg-rr 38.2%

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

   7. Taylor expanded in c0 around inf 47.1%

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

      1. associate-/l* 47.1%

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

      2. associate-/r* 47.1%

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

      3. unpow2 47.1%

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

      4. unpow2 47.1%

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

      5. times-frac 56.7%

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

      6. unpow2 56.7%

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

   9. Simplified 56.7%

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

       1. unpow2 56.7%

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

       2. *-commutative 56.7%

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

       3. times-frac 59.8%

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

   11. Applied egg-rr 59.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9 \cdot 10^{+223} \lor \neg \left(d \leq 4.4 \cdot 10^{+295}\right):\\ \;\;\;\;0.25 \cdot \left(h \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \left(c0 \cdot \left(\frac{\frac{d}{D}}{w} \cdot \frac{\frac{d}{D}}{h}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.2% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{D \cdot M}{d}\\ 0.25 \cdot \left(h \cdot \left(t\_0 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* D M) d))) (* 0.25 (* h (* t_0 t_0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * M) / d;
	return 0.25 * (h * (t_0 * t_0));
}

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 = (d * m) / d_1
    code = 0.25d0 * (h * (t_0 * t_0))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (D * M) / d;
	return 0.25 * (h * (t_0 * t_0));
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (D * M) / d
	return 0.25 * (h * (t_0 * t_0))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(D * M) / d)
	return Float64(0.25 * Float64(h * Float64(t_0 * t_0)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (D * M) / d;
	tmp = 0.25 * (h * (t_0 * t_0));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(D * M), $MachinePrecision] / d), $MachinePrecision]}, N[(0.25 * N[(h * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 28.7%

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

3. Simplified 28.0%

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

5. Taylor expanded in c0 around -inf 6.3%

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

   1. +-commutative 6.3%

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

   2. associate-/l* 6.6%

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

   3. *-commutative 6.6%

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

   4. distribute-lft-in 5.4%

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

7. Simplified 25.9%

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

8. Taylor expanded in c0 around 0 37.3%

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

   1. associate-/l* 37.3%

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

   2. pow2 37.3%

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

   3. *-commutative 37.3%

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

   4. pow2 37.3%

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

10. Applied egg-rr 37.3%

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

    1. associate-*r/ 37.3%

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

    2. *-commutative 37.3%

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

    3. associate-*r* 38.3%

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

    4. unpow2 38.3%

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

    5. unpow2 38.3%

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

    6. swap-sqr 44.7%

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

    7. unpow2 44.7%

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

    8. *-commutative 44.7%

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

    9. associate-/l* 44.8%

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

    10. *-commutative 44.8%

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

12. Simplified 44.8%

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

    1. unpow2 44.8%

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

    2. unpow2 44.8%

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

    3. times-frac 53.3%

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

14. Applied egg-rr 53.3%

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

15. Final simplification 53.3%

    \[\leadsto 0.25 \cdot \left(h \cdot \left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right)\right) \]
16. Add Preprocessing

Alternative 4: 34.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ c0 \cdot \frac{0}{2 \cdot w} \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* c0 (/ 0.0 (* 2.0 w))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Fortran

real(8) function code(c0, w, h, d, d_1, m)
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = c0 * (0.0d0 / (2.0d0 * w))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (0.0 / (2.0 * w));
}

Python

def code(c0, w, h, D, d, M):
	return c0 * (0.0 / (2.0 * w))

Julia

function code(c0, w, h, D, d, M)
	return Float64(c0 * Float64(0.0 / Float64(2.0 * w)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = c0 * (0.0 / (2.0 * w));
end

Wolfram

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

Derivation

1. Initial program 28.7%

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

3. Simplified 39.3%

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

5. Taylor expanded in c0 around -inf 3.9%

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

   1. distribute-lft-in 3.1%

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

   2. mul-1-neg 3.1%

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

   3. distribute-rgt-neg-in 3.1%

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

   4. associate-/l* 2.7%

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

   5. mul-1-neg 2.7%

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

   6. associate-/l* 3.1%

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

   7. distribute-lft1-in 3.1%

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

   8. metadata-eval 3.1%

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

   9. mul0-lft 34.3%

      \[\leadsto c0 \cdot \frac{-1 \cdot \color{blue}{0}}{2 \cdot w} \]

   10. metadata-eval 34.3%

       \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

7. Simplified 34.3%

   \[\leadsto c0 \cdot \frac{\color{blue}{0}}{2 \cdot w} \]

8. Final simplification 34.3%

   \[\leadsto c0 \cdot \frac{0}{2 \cdot w} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))))