Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 63.5%

Time: 16.4s

Alternatives: 6

Speedup: 156.0×

• could not determine a ground truth

  1. c0 = -1.8946917962188417e+155
  2. w = 2.4544288224877699e-265
  3. h = 2.46356924388415e-294
  4. D = 5.873301141213449e-293
  5. d = -2.3693335954527903e+295
  6. M = 4.4167841045298146e+189

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: 24.4% 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: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(d \cdot \left(c0 \cdot d\right)\right)}{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(0.25 \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{d \cdot d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (/ (* 2.0 (* d (* c0 d))) (* D (* D (* w h)))))
     (* D (* 0.25 (* D (* M (* h (/ M (* d d))))))))))

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)) / ((D * D) * (w * h));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * (d * (c0 * d))) / (D * (D * (w * h))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	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)) / ((D * D) * (w * h));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * (d * (c0 * d))) / (D * (D * (w * h))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * ((2.0 * (d * (c0 * d))) / (D * (D * (w * h))))
	else:
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))))
	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(D * D) * Float64(w * h)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(d * Float64(c0 * d))) / Float64(D * Float64(D * Float64(w * h)))));
	else
		tmp = Float64(D * Float64(0.25 * Float64(D * Float64(M * Float64(h * Float64(M / Float64(d * d)))))));
	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)) / ((D * D) * (w * h));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * (d * (c0 * d))) / (D * (D * (w * h))));
	else
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	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[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(d * N[(c0 * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D * N[(0.25 * N[(D * N[(M * N[(h * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 69.8

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

   8. Simplified 69.8%

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

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 76.6

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

   11. Simplified 76.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 22.4%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}}{{d}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}{{d}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)} \cdot \frac{1}{4}}{{d}^{2}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{D \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      10. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      14. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      16. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}\right)}{\color{blue}{d \cdot d}} \]

      17. lower-*.f64 45.0

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

   8. Simplified 45.0%

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

   9. Taylor expanded in D around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left(D \cdot {M}^{2}\right) \cdot h\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       2. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       5. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       6. lower-*.f64 47.2

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

   11. Simplified 47.2%

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

   12. Taylor expanded in D around 0

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

       1. associate-/l* N/A

          \[\leadsto \frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

       9. associate-*l* N/A

          \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       12. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       16. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\frac{M \cdot h}{{d}^{2}}}\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{{d}^{2}}\right)\right)\right) \]

       18. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       20. lower-/.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{{d}^{2}}}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{d \cdot d}}\right)\right)\right)\right) \]

       22. lower-*.f64 52.8

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

   14. Simplified 52.8%

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

4. Final simplification 59.3%

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

Alternative 2: 63.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* c0 (* d d)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* D D) (* w h)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (/ (* 2.0 t_0) (* D (* D (* w h)))))
     (* D (* 0.25 (* D (* M (* h (/ M (* d d))))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * (d * d);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((D * D) * (w * h));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((2.0 * t_0) / (D * (D * (w * h))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	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);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((D * D) * (w * h));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((2.0 * t_0) / (D * (D * (w * h))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * (d * d)
	t_1 = c0 / (2.0 * w)
	t_2 = t_0 / ((D * D) * (w * h))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * ((2.0 * t_0) / (D * (D * (w * h))))
	else:
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(D * D), $MachinePrecision] * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(2.0 * t$95$0), $MachinePrecision] / N[(D * N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D * N[(0.25 * N[(D * N[(M * N[(h * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 69.8

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

   8. Simplified 69.8%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 22.4%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}}{{d}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}{{d}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)} \cdot \frac{1}{4}}{{d}^{2}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{D \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      10. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      14. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      16. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}\right)}{\color{blue}{d \cdot d}} \]

      17. lower-*.f64 45.0

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

   8. Simplified 45.0%

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

   9. Taylor expanded in D around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left(D \cdot {M}^{2}\right) \cdot h\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       2. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       5. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       6. lower-*.f64 47.2

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

   11. Simplified 47.2%

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

   12. Taylor expanded in D around 0

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

       1. associate-/l* N/A

          \[\leadsto \frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

       9. associate-*l* N/A

          \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       12. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       16. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\frac{M \cdot h}{{d}^{2}}}\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{{d}^{2}}\right)\right)\right) \]

       18. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       20. lower-/.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{{d}^{2}}}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{d \cdot d}}\right)\right)\right)\right) \]

       22. lower-*.f64 52.8

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

   14. Simplified 52.8%

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

4. Final simplification 57.4%

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

Alternative 3: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(D \cdot \left(w \cdot \left(w \cdot h\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(0.25 \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{d \cdot d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* d (* d (* c0 c0))) (* D (* D (* w (* w h)))))
     (* D (* 0.25 (* D (* M (* h (/ M (* d d))))))))))

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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (d * (d * (c0 * c0))) / (D * (D * (w * (w * h))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (d * (d * (c0 * c0))) / (D * (D * (w * (w * h))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (d * (d * (c0 * c0))) / (D * (D * (w * (w * h))))
	else:
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))))
	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)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(D * Float64(D * Float64(w * Float64(w * h)))));
	else
		tmp = Float64(D * Float64(0.25 * Float64(D * Float64(M * Float64(h * Float64(M / Float64(d * d)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (d * (d * (c0 * c0))) / (D * (D * (w * (w * h))));
	else
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	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]}, If[LessEqual[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], Infinity], N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(w * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D * N[(0.25 * N[(D * N[(M * N[(h * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 69.8

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

   8. Simplified 69.8%

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

   9. Taylor expanded in c0 around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 60.4

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

   11. Simplified 60.4%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 22.4%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}}{{d}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}{{d}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)} \cdot \frac{1}{4}}{{d}^{2}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{D \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      10. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      14. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      16. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}\right)}{\color{blue}{d \cdot d}} \]

      17. lower-*.f64 45.0

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

   8. Simplified 45.0%

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

   9. Taylor expanded in D around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left(D \cdot {M}^{2}\right) \cdot h\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       2. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       5. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       6. lower-*.f64 47.2

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

   11. Simplified 47.2%

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

   12. Taylor expanded in D around 0

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

       1. associate-/l* N/A

          \[\leadsto \frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

       9. associate-*l* N/A

          \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       12. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       16. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\frac{M \cdot h}{{d}^{2}}}\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{{d}^{2}}\right)\right)\right) \]

       18. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       20. lower-/.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{{d}^{2}}}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{d \cdot d}}\right)\right)\right)\right) \]

       22. lower-*.f64 52.8

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

   14. Simplified 52.8%

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

4. Final simplification 54.9%

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

Alternative 4: 57.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{d \cdot \left(d \cdot \left(c0 \cdot c0\right)\right)}{D \cdot \left(D \cdot \left(h \cdot \left(w \cdot w\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;D \cdot \left(0.25 \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{d \cdot d}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* D D) (* w h)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M M)))))
        INFINITY)
     (/ (* d (* d (* c0 c0))) (* D (* D (* h (* w w)))))
     (* D (* 0.25 (* D (* M (* h (/ M (* d d))))))))))

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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = (d * (d * (c0 * c0))) / (D * (D * (h * (w * w))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	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 tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (d * (d * (c0 * c0))) / (D * (D * (h * (w * w))));
	} else {
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (d * (d * (c0 * c0))) / (D * (D * (h * (w * w))))
	else:
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))))
	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)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(d * Float64(d * Float64(c0 * c0))) / Float64(D * Float64(D * Float64(h * Float64(w * w)))));
	else
		tmp = Float64(D * Float64(0.25 * Float64(D * Float64(M * Float64(h * Float64(M / Float64(d * d)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((D * D) * (w * h));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (d * (d * (c0 * c0))) / (D * (D * (h * (w * w))));
	else
		tmp = D * (0.25 * (D * (M * (h * (M / (d * d))))));
	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]}, If[LessEqual[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], Infinity], N[(N[(d * N[(d * N[(c0 * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * N[(D * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(D * N[(0.25 * N[(D * N[(M * N[(h * N[(M / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 72.6%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 69.8

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

   8. Simplified 69.8%

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

   9. Taylor expanded in c0 around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. unpow2 N/A

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

       15. associate-*l* N/A

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

       16. *-commutative N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 60.4

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

   11. Simplified 60.4%

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

   12. Taylor expanded in d around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. associate-*l* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 56.6

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

   14. Simplified 56.6%

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

   if +inf.0 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

   5. Simplified 22.4%

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

   6. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}}{{d}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}{{d}^{2}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)} \cdot \frac{1}{4}}{{d}^{2}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{D \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

      10. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      14. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

      16. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}\right)}{\color{blue}{d \cdot d}} \]

      17. lower-*.f64 45.0

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

   8. Simplified 45.0%

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

   9. Taylor expanded in D around 0

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

       1. associate-*r* N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left(D \cdot {M}^{2}\right) \cdot h\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       2. *-commutative N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       5. unpow2 N/A

          \[\leadsto \frac{D \cdot \left(\left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

       6. lower-*.f64 47.2

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

   11. Simplified 47.2%

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

   12. Taylor expanded in D around 0

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

       1. associate-/l* N/A

          \[\leadsto \frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

       3. *-commutative N/A

          \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       4. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

       8. *-commutative N/A

          \[\leadsto D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

       9. associate-*l* N/A

          \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

       12. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]

       13. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

       16. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\frac{M \cdot h}{{d}^{2}}}\right)\right)\right) \]

       17. *-commutative N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{{d}^{2}}\right)\right)\right) \]

       18. associate-/l* N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       19. lower-*.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

       20. lower-/.f64 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{{d}^{2}}}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{d \cdot d}}\right)\right)\right)\right) \]

       22. lower-*.f64 52.8

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

   14. Simplified 52.8%

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

4. Final simplification 53.8%

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

Alternative 5: 44.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

3. Taylor expanded in c0 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

5. Simplified 19.4%

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

6. Taylor expanded in c0 around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({D}^{2} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}}{{d}^{2}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\left(\color{blue}{\left(D \cdot D\right)} \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}}{{d}^{2}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(D \cdot \left(D \cdot \left({M}^{2} \cdot h\right)\right)\right)} \cdot \frac{1}{4}}{{d}^{2}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{D \cdot \left(\left(D \cdot \left({M}^{2} \cdot h\right)\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

   8. *-commutative N/A

      \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left({M}^{2} \cdot h\right) \cdot D\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{D \cdot \color{blue}{\left(\left(\left({M}^{2} \cdot h\right) \cdot D\right) \cdot \frac{1}{4}\right)}}{{d}^{2}} \]

   10. *-commutative N/A

       \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{D \cdot \left(\color{blue}{\left(D \cdot \left({M}^{2} \cdot h\right)\right)} \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   12. *-commutative N/A

       \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{D \cdot \left(\left(D \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   14. unpow2 N/A

       \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{{d}^{2}} \]

   16. unpow2 N/A

       \[\leadsto \frac{D \cdot \left(\left(D \cdot \left(h \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{1}{4}\right)}{\color{blue}{d \cdot d}} \]

   17. lower-*.f64 38.2

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

8. Simplified 38.2%

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

9. Taylor expanded in D around 0

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

    1. associate-*r* N/A

       \[\leadsto \frac{D \cdot \left(\color{blue}{\left(\left(D \cdot {M}^{2}\right) \cdot h\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

    2. *-commutative N/A

       \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{D \cdot \left(\color{blue}{\left(h \cdot \left(D \cdot {M}^{2}\right)\right)} \cdot \frac{1}{4}\right)}{d \cdot d} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{D \cdot \left(\left(h \cdot \color{blue}{\left(D \cdot {M}^{2}\right)}\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

    5. unpow2 N/A

       \[\leadsto \frac{D \cdot \left(\left(h \cdot \left(D \cdot \color{blue}{\left(M \cdot M\right)}\right)\right) \cdot \frac{1}{4}\right)}{d \cdot d} \]

    6. lower-*.f64 39.9

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

11. Simplified 39.9%

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

12. Taylor expanded in D around 0

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

    1. associate-/l* N/A

       \[\leadsto \frac{1}{4} \cdot \color{blue}{\left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot {D}^{2}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{\left({D}^{2} \cdot \frac{1}{4}\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

    4. unpow2 N/A

       \[\leadsto \left(\color{blue}{\left(D \cdot D\right)} \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

    5. associate-*l* N/A

       \[\leadsto \color{blue}{\left(D \cdot \left(D \cdot \frac{1}{4}\right)\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} \]

    6. associate-*l* N/A

       \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \color{blue}{D \cdot \left(\left(D \cdot \frac{1}{4}\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)} \]

    8. *-commutative N/A

       \[\leadsto D \cdot \left(\color{blue}{\left(\frac{1}{4} \cdot D\right)} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \]

    9. associate-*l* N/A

       \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

    10. lower-*.f64 N/A

        \[\leadsto D \cdot \color{blue}{\left(\frac{1}{4} \cdot \left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)\right)} \]

    11. lower-*.f64 N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \color{blue}{\left(D \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right)}\right) \]

    12. associate-/l* N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left({M}^{2} \cdot \frac{h}{{d}^{2}}\right)}\right)\right) \]

    13. unpow2 N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{h}{{d}^{2}}\right)\right)\right) \]

    14. associate-*l* N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

    15. lower-*.f64 N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \color{blue}{\left(M \cdot \left(M \cdot \frac{h}{{d}^{2}}\right)\right)}\right)\right) \]

    16. associate-/l* N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\frac{M \cdot h}{{d}^{2}}}\right)\right)\right) \]

    17. *-commutative N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \frac{\color{blue}{h \cdot M}}{{d}^{2}}\right)\right)\right) \]

    18. associate-/l* N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

    19. lower-*.f64 N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \color{blue}{\left(h \cdot \frac{M}{{d}^{2}}\right)}\right)\right)\right) \]

    20. lower-/.f64 N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \color{blue}{\frac{M}{{d}^{2}}}\right)\right)\right)\right) \]

    21. unpow2 N/A

        \[\leadsto D \cdot \left(\frac{1}{4} \cdot \left(D \cdot \left(M \cdot \left(h \cdot \frac{M}{\color{blue}{d \cdot d}}\right)\right)\right)\right) \]

    22. lower-*.f64 45.1

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

14. Simplified 45.1%

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

Alternative 6: 33.5% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

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

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
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 19.9%

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

3. Taylor expanded in c0 around -inf

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

   1. associate-*r* N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

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

   5. metadata-eval N/A

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

   6. mul-1-neg N/A

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

   7. distribute-lft-neg-in N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval 27.3

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

5. Simplified 27.3%

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

6. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 32.4%

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

Reproduce

herbie shell --seed 2024215 
(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))))))