Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.4% → 52.2%

Time: 7.5s

Alternatives: 14

Speedup: 2.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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: 52.2% accurate, 0.5× speedup

Math

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

FPCore

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

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	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(c0 / Float64(w + w)) * Float64(Float64(2.0 * Float64(Float64(d * c0) * d)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(Float64((Float64(D * M) ^ 2.0) * h) / Float64(d * d)) * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 / (w + w)) * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	else
		tmp = ((((D * M) ^ 2.0) * h) / (d * d)) * -0.25;
	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[(w * h), $MachinePrecision] * N[(D * D), $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[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(D * M), $MachinePrecision], 2.0], $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * -0.25), $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 73.7%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 77.6

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

   9. Applied rewrites 77.6%

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

   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 D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

4. Final simplification 50.8%

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

Alternative 2: 39.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.12 \cdot 10^{-75}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{h \cdot \left(w \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}\\ \mathbf{elif}\;D \leq 4.1 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot \frac{c0}{\left(h \cdot w\right) \cdot D}}{D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 1.12e-75)
   (/ (* c0 (* (/ 2.0 (* h (* w D))) (/ (* (* d d) c0) D))) (* w 2.0))
   (if (<= D 4.1e+142)
     (/ (/ (/ (* (* d c0) (* d c0)) h) (* w w)) (* D D))
     (* (/ c0 (* 2.0 w)) (/ (* (* 2.0 (* d d)) (/ c0 (* (* h w) D))) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.12e-75) {
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0);
	} else if (D <= 4.1e+142) {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	} else {
		tmp = (c0 / (2.0 * w)) * (((2.0 * (d * d)) * (c0 / ((h * w) * D))) / D);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 1.12d-75) then
        tmp = (c0 * ((2.0d0 / (h * (w * d))) * (((d_1 * d_1) * c0) / d))) / (w * 2.0d0)
    else if (d <= 4.1d+142) then
        tmp = ((((d_1 * c0) * (d_1 * c0)) / h) / (w * w)) / (d * d)
    else
        tmp = (c0 / (2.0d0 * w)) * (((2.0d0 * (d_1 * d_1)) * (c0 / ((h * w) * d))) / d)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.12e-75) {
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0);
	} else if (D <= 4.1e+142) {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	} else {
		tmp = (c0 / (2.0 * w)) * (((2.0 * (d * d)) * (c0 / ((h * w) * D))) / D);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 1.12e-75:
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0)
	elif D <= 4.1e+142:
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D)
	else:
		tmp = (c0 / (2.0 * w)) * (((2.0 * (d * d)) * (c0 / ((h * w) * D))) / D)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 1.12e-75)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(h * Float64(w * D))) * Float64(Float64(Float64(d * d) * c0) / D))) / Float64(w * 2.0));
	elseif (D <= 4.1e+142)
		tmp = Float64(Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / h) / Float64(w * w)) / Float64(D * D));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(2.0 * Float64(d * d)) * Float64(c0 / Float64(Float64(h * w) * D))) / D));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 1.12e-75)
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0);
	elseif (D <= 4.1e+142)
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	else
		tmp = (c0 / (2.0 * w)) * (((2.0 * (d * d)) * (c0 / ((h * w) * D))) / D);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 1.12e-75], N[(N[(c0 * N[(N[(2.0 / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 4.1e+142], N[(N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * N[(c0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if D < 1.12000000000000004e-75

   1. Initial program 24.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 38.1

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

   5. Applied rewrites 38.1%

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

   7. Applied rewrites 38.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 42.0

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

   9. Applied rewrites 42.0%

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

   if 1.12000000000000004e-75 < D < 4.09999999999999982e142

   1. Initial program 20.6%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 19.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 41.7

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

   8. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 41.7

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

   10. Applied rewrites 41.7%

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

   if 4.09999999999999982e142 < D

   1. Initial program 0.2%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 23.3

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

   7. Applied rewrites 23.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 23.6

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

   9. Applied rewrites 23.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 1.12 \cdot 10^{-75}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{h \cdot \left(w \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}\\ \mathbf{elif}\;D \leq 4.1 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot \frac{c0}{\left(h \cdot w\right) \cdot D}}{D}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 8 \cdot 10^{-111}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{\frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{h \cdot w}}{D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{c0}{D} \cdot \left(\frac{d}{h \cdot w} \cdot \frac{d}{D}\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 8e-111)
   (* (/ c0 (+ w w)) (/ (/ (/ (* (* 2.0 (* d d)) c0) (* h w)) D) D))
   (* (/ c0 (* 2.0 w)) (* (* (/ c0 D) (* (/ d (* h w)) (/ d D))) 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 8e-111) {
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D);
	} else {
		tmp = (c0 / (2.0 * w)) * (((c0 / D) * ((d / (h * w)) * (d / D))) * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 8d-111) then
        tmp = (c0 / (w + w)) * (((((2.0d0 * (d_1 * d_1)) * c0) / (h * w)) / d) / d)
    else
        tmp = (c0 / (2.0d0 * w)) * (((c0 / d) * ((d_1 / (h * w)) * (d_1 / d))) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 8e-111) {
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D);
	} else {
		tmp = (c0 / (2.0 * w)) * (((c0 / D) * ((d / (h * w)) * (d / D))) * 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 8e-111:
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D)
	else:
		tmp = (c0 / (2.0 * w)) * (((c0 / D) * ((d / (h * w)) * (d / D))) * 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 8e-111)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(Float64(Float64(Float64(2.0 * Float64(d * d)) * c0) / Float64(h * w)) / D) / D));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 / D) * Float64(Float64(d / Float64(h * w)) * Float64(d / D))) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 8e-111)
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D);
	else
		tmp = (c0 / (2.0 * w)) * (((c0 / D) * ((d / (h * w)) * (d / D))) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 8e-111], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 / D), $MachinePrecision] * N[(N[(d / N[(h * w), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 8.00000000000000071e-111

   1. Initial program 23.7%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 38.0

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

   5. Applied rewrites 38.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 40.6

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

   7. Applied rewrites 40.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 42.7

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

   9. Applied rewrites 42.7%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lift-+.f64 42.7

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

   11. Applied rewrites 42.7%

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

   if 8.00000000000000071e-111 < D

   1. Initial program 16.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 31.6

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

   7. Applied rewrites 31.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 31.6

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

   9. Applied rewrites 31.6%

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

   11. Applied rewrites 36.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 8 \cdot 10^{-111}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{\frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{h \cdot w}}{D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{c0}{D} \cdot \left(\frac{d}{h \cdot w} \cdot \frac{d}{D}\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 5.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{\frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{h \cdot w}}{D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{d \cdot \left(d \cdot c0\right)}{D}\right)}{w \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 5.4e-77)
   (* (/ c0 (+ w w)) (/ (/ (/ (* (* 2.0 (* d d)) c0) (* h w)) D) D))
   (/ (* c0 (* (/ 2.0 (* (* h w) D)) (/ (* d (* d c0)) D))) (* w 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 5.4e-77) {
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D);
	} else {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 5.4d-77) then
        tmp = (c0 / (w + w)) * (((((2.0d0 * (d_1 * d_1)) * c0) / (h * w)) / d) / d)
    else
        tmp = (c0 * ((2.0d0 / ((h * w) * d)) * ((d_1 * (d_1 * c0)) / d))) / (w * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 5.4e-77) {
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D);
	} else {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 5.4e-77:
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D)
	else:
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 5.4e-77)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(Float64(Float64(Float64(2.0 * Float64(d * d)) * c0) / Float64(h * w)) / D) / D));
	else
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(Float64(h * w) * D)) * Float64(Float64(d * Float64(d * c0)) / D))) / Float64(w * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 5.4e-77)
		tmp = (c0 / (w + w)) * (((((2.0 * (d * d)) * c0) / (h * w)) / D) / D);
	else
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 5.4e-77], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[(h * w), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[(N[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5.4000000000000001e-77

   1. Initial program 23.6%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 37.8

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

   5. Applied rewrites 37.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 40.3

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

   7. Applied rewrites 40.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 42.4

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

   9. Applied rewrites 42.4%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lift-+.f64 42.4

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

   11. Applied rewrites 42.4%

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

   if 5.4000000000000001e-77 < D

   1. Initial program 15.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 28.7

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

   5. Applied rewrites 28.7%

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

   7. Applied rewrites 31.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.2

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

   9. Applied rewrites 33.2%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 5.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{\frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{h \cdot w}}{D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{d \cdot \left(d \cdot c0\right)}{D}\right)}{w \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{h \cdot \left(w \cdot D\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\frac{2}{\left(h \cdot w\right) \cdot D} \cdot \frac{d \cdot \left(d \cdot c0\right)}{D}\right)}{w \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 5e-108)
   (/ (* c0 (* (/ 2.0 (* h (* w D))) (/ (* (* d d) c0) D))) (* w 2.0))
   (/ (* c0 (* (/ 2.0 (* (* h w) D)) (/ (* d (* d c0)) D))) (* w 2.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 5e-108) {
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0);
	} else {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d <= 5d-108) then
        tmp = (c0 * ((2.0d0 / (h * (w * d))) * (((d_1 * d_1) * c0) / d))) / (w * 2.0d0)
    else
        tmp = (c0 * ((2.0d0 / ((h * w) * d)) * ((d_1 * (d_1 * c0)) / d))) / (w * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 5e-108) {
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0);
	} else {
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 5e-108:
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0)
	else:
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 5e-108)
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(h * Float64(w * D))) * Float64(Float64(Float64(d * d) * c0) / D))) / Float64(w * 2.0));
	else
		tmp = Float64(Float64(c0 * Float64(Float64(2.0 / Float64(Float64(h * w) * D)) * Float64(Float64(d * Float64(d * c0)) / D))) / Float64(w * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 5e-108)
		tmp = (c0 * ((2.0 / (h * (w * D))) * (((d * d) * c0) / D))) / (w * 2.0);
	else
		tmp = (c0 * ((2.0 / ((h * w) * D)) * ((d * (d * c0)) / D))) / (w * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 5e-108], N[(N[(c0 * N[(N[(2.0 / N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[(N[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 5e-108

   1. Initial program 23.7%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 38.0

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

   5. Applied rewrites 38.0%

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

   7. Applied rewrites 38.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 42.0

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

   9. Applied rewrites 42.0%

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

   if 5e-108 < D

   1. Initial program 16.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

   7. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.4

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

   9. Applied rewrites 33.4%

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

Alternative 6: 37.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.9 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\left(\left(d \cdot d\right) \cdot 2\right) \cdot \frac{c0}{\left(D \cdot h\right) \cdot w}}{D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 5.9e-159)
   (/ (/ (/ (* (* d c0) (* d c0)) h) (* w w)) (* D D))
   (* (/ c0 (* 2.0 w)) (/ (* (* (* d d) 2.0) (/ c0 (* (* D h) w))) D))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 5.9e-159) {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	} else {
		tmp = (c0 / (2.0 * w)) * ((((d * d) * 2.0) * (c0 / ((D * h) * w))) / D);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 5.9d-159) then
        tmp = ((((d_1 * c0) * (d_1 * c0)) / h) / (w * w)) / (d * d)
    else
        tmp = (c0 / (2.0d0 * w)) * ((((d_1 * d_1) * 2.0d0) * (c0 / ((d * h) * w))) / d)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 5.9e-159) {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	} else {
		tmp = (c0 / (2.0 * w)) * ((((d * d) * 2.0) * (c0 / ((D * h) * w))) / D);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 5.9e-159:
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D)
	else:
		tmp = (c0 / (2.0 * w)) * ((((d * d) * 2.0) * (c0 / ((D * h) * w))) / D)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 5.9e-159)
		tmp = Float64(Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / h) / Float64(w * w)) / Float64(D * D));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(Float64(d * d) * 2.0) * Float64(c0 / Float64(Float64(D * h) * w))) / D));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 5.9e-159)
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	else
		tmp = (c0 / (2.0 * w)) * ((((d * d) * 2.0) * (c0 / ((D * h) * w))) / D);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 5.9e-159], N[(N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(d * d), $MachinePrecision] * 2.0), $MachinePrecision] * N[(c0 / N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.9e-159

   1. Initial program 18.1%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 18.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 35.2

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

   10. Applied rewrites 35.2%

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

   if 5.9e-159 < d

   1. Initial program 26.2%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 36.9

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

   7. Applied rewrites 36.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 38.5

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

   9. Applied rewrites 38.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/l* N/A

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

       11. lower-*.f64 N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. pow2 N/A

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

       16. lift-*.f64 N/A

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

       17. lower-/.f64 35.9

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

       18. lift-*.f64 N/A

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

       19. lift-*.f64 N/A

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

       20. *-commutative N/A

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

       21. associate-*r* N/A

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

       22. lower-*.f64 N/A

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

       23. lower-*.f64 39.7

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

   11. Applied rewrites 39.7%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.9 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\left(\left(d \cdot d\right) \cdot 2\right) \cdot \frac{c0}{\left(D \cdot h\right) \cdot w}}{D}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 6.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{2}{\left(D \cdot h\right) \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= d 6.3e-158)
   (/ (/ (/ (* (* d c0) (* d c0)) h) (* w w)) (* D D))
   (* (/ c0 (* 2.0 w)) (* (/ 2.0 (* (* D h) w)) (/ (* (* d d) c0) D)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 6.3e-158) {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 / ((D * h) * w)) * (((d * d) * c0) / D));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (d_1 <= 6.3d-158) then
        tmp = ((((d_1 * c0) * (d_1 * c0)) / h) / (w * w)) / (d * d)
    else
        tmp = (c0 / (2.0d0 * w)) * ((2.0d0 / ((d * h) * w)) * (((d_1 * d_1) * c0) / d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (d <= 6.3e-158) {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	} else {
		tmp = (c0 / (2.0 * w)) * ((2.0 / ((D * h) * w)) * (((d * d) * c0) / D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if d <= 6.3e-158:
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D)
	else:
		tmp = (c0 / (2.0 * w)) * ((2.0 / ((D * h) * w)) * (((d * d) * c0) / D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (d <= 6.3e-158)
		tmp = Float64(Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / h) / Float64(w * w)) / Float64(D * D));
	else
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(2.0 / Float64(Float64(D * h) * w)) * Float64(Float64(Float64(d * d) * c0) / D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (d <= 6.3e-158)
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	else
		tmp = (c0 / (2.0 * w)) * ((2.0 / ((D * h) * w)) * (((d * d) * c0) / D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[d, 6.3e-158], N[(N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[(N[(D * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.30000000000000014e-158

   1. Initial program 18.1%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 18.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 35.2

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

   10. Applied rewrites 35.2%

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

   if 6.30000000000000014e-158 < d

   1. Initial program 26.2%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 36.9

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

   7. Applied rewrites 36.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 38.5

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

   9. Applied rewrites 38.5%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/l/ N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. associate-*r* N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. times-frac N/A

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

   11. Applied rewrites 40.0%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\frac{2}{\left(D \cdot h\right) \cdot w} \cdot \frac{\left(d \cdot d\right) \cdot c0}{D}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c0 \leq 2.2 \cdot 10^{+188}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{\left(h \cdot w\right) \cdot D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= c0 2.2e+188)
   (* (/ c0 (+ w w)) (/ (/ (* (* 2.0 (* d d)) c0) (* (* h w) D)) D))
   (/ (/ (/ (* (* d c0) (* d c0)) h) (* w w)) (* D D))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= 2.2e+188) {
		tmp = (c0 / (w + w)) * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D);
	} else {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if (c0 <= 2.2d+188) then
        tmp = (c0 / (w + w)) * ((((2.0d0 * (d_1 * d_1)) * c0) / ((h * w) * d)) / d)
    else
        tmp = ((((d_1 * c0) * (d_1 * c0)) / h) / (w * w)) / (d * d)
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (c0 <= 2.2e+188) {
		tmp = (c0 / (w + w)) * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D);
	} else {
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if c0 <= 2.2e+188:
		tmp = (c0 / (w + w)) * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D)
	else:
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (c0 <= 2.2e+188)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(Float64(Float64(2.0 * Float64(d * d)) * c0) / Float64(Float64(h * w) * D)) / D));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / h) / Float64(w * w)) / Float64(D * D));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (c0 <= 2.2e+188)
		tmp = (c0 / (w + w)) * ((((2.0 * (d * d)) * c0) / ((h * w) * D)) / D);
	else
		tmp = ((((d * c0) * (d * c0)) / h) / (w * w)) / (D * D);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[c0, 2.2e+188], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c0 < 2.19999999999999999e188

   1. Initial program 22.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 35.9

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

   5. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 38.7

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

   7. Applied rewrites 38.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 38.7

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

   9. Applied rewrites 38.7%

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

   if 2.19999999999999999e188 < c0

   1. Initial program 16.7%

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

   3. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 13.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 54.3

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

   8. Applied rewrites 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 54.3

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

   10. Applied rewrites 54.3%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \leq 2.2 \cdot 10^{+188}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{\frac{\left(2 \cdot \left(d \cdot d\right)\right) \cdot c0}{\left(h \cdot w\right) \cdot D}}{D}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{w + w}\\ \mathbf{if}\;D \leq 7.6 \cdot 10^{-110}:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (+ w w))))
   (if (<= D 7.6e-110)
     (* t_0 (/ (* 2.0 (* (* d d) c0)) (* (* h (* w D)) D)))
     (* t_0 (/ (* 2.0 (* (* d c0) d)) (* (* (* h w) D) D))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w + w);
	double tmp;
	if (D <= 7.6e-110) {
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D));
	} else {
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 / (w + w)
    if (d <= 7.6d-110) then
        tmp = t_0 * ((2.0d0 * ((d_1 * d_1) * c0)) / ((h * (w * d)) * d))
    else
        tmp = t_0 * ((2.0d0 * ((d_1 * c0) * d_1)) / (((h * w) * d) * d))
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (w + w);
	double tmp;
	if (D <= 7.6e-110) {
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D));
	} else {
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (w + w)
	tmp = 0
	if D <= 7.6e-110:
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D))
	else:
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(w + w))
	tmp = 0.0
	if (D <= 7.6e-110)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(Float64(d * d) * c0)) / Float64(Float64(h * Float64(w * D)) * D)));
	else
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(Float64(d * c0) * d)) / Float64(Float64(Float64(h * w) * D) * D)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (w + w);
	tmp = 0.0;
	if (D <= 7.6e-110)
		tmp = t_0 * ((2.0 * ((d * d) * c0)) / ((h * (w * D)) * D));
	else
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, 7.6e-110], N[(t$95$0 * N[(N[(2.0 * N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(2.0 * N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 7.5999999999999996e-110

   1. Initial program 23.7%

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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 38.0

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

   5. Applied rewrites 38.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 38.0

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

   7. Applied rewrites 38.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 40.6

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

   9. Applied rewrites 40.6%

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

   if 7.5999999999999996e-110 < D

   1. Initial program 16.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 29.6

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

   5. Applied rewrites 29.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 29.6

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

   7. Applied rewrites 29.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 31.5

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

   9. Applied rewrites 31.5%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq 7.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.1

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

5. Applied rewrites 36.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 36.1

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

7. Applied rewrites 36.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. *-commutative N/A

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

   5. pow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 37.8

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

9. Applied rewrites 37.8%

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

10. Final simplification 37.8%

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

Alternative 11: 33.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    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_1 * c0) * (d_1 * c0)) / h) / (w * w)) / (d * d)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in D around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

6. Taylor expanded in c0 around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow-prod-down N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 33.5

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

8. Applied rewrites 33.5%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 33.5

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

10. Applied rewrites 33.5%

    \[\leadsto \frac{\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{h}}{w \cdot w}}{D \cdot D} \]
11. Add Preprocessing

Alternative 12: 26.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c0 \cdot c0}{h} \cdot \frac{d \cdot d}{w \cdot w}}{D \cdot D} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in D around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

6. Taylor expanded in c0 around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow-prod-down N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 33.5

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

8. Applied rewrites 33.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. pow2 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. associate-/l/ N/A

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

   8. unpow-prod-down N/A

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

   9. *-commutative N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 N/A

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

   18. pow2 N/A

       \[\leadsto \frac{\frac{c0 \cdot c0}{h} \cdot \frac{d \cdot d}{w \cdot w}}{D \cdot D} \]

   19. lift-*.f64 26.7

       \[\leadsto \frac{\frac{c0 \cdot c0}{h} \cdot \frac{d \cdot d}{w \cdot w}}{D \cdot D} \]

10. Applied rewrites 26.7%

    \[\leadsto \frac{\frac{c0 \cdot c0}{h} \cdot \frac{d \cdot d}{w \cdot w}}{D \cdot D} \]
11. Add Preprocessing

Alternative 13: 25.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{d \cdot d}{h}}{w \cdot w}}{D \cdot D} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in D around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{4} \cdot \frac{{D}^{4} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}}{{D}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{4} \cdot \frac{{D}^{4} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} + \frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}}{\color{blue}{{D}^{2}}} \]

5. Applied rewrites 18.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{4}}{d \cdot d}, -0.25, \frac{{\left(c0 \cdot d\right)}^{2}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

6. Taylor expanded in c0 around inf

   \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}}{\color{blue}{D} \cdot D} \]
7. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{{c0}^{2} \cdot {d}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{{c0}^{2} \cdot {d}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{{c0}^{2} \cdot {d}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   4. unpow-prod-down N/A

      \[\leadsto \frac{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   6. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{w \cdot w}}{D \cdot D} \]

   9. lift-*.f64 33.5

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{w \cdot w}}{D \cdot D} \]

8. Applied rewrites 33.5%

   \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{w \cdot w}}{\color{blue}{D} \cdot D} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{w \cdot w}}{D \cdot D} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{w \cdot w}}{D \cdot D} \]

   3. pow2 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\frac{{\left(d \cdot c0\right)}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   7. associate-/l/ N/A

      \[\leadsto \frac{\frac{{\left(d \cdot c0\right)}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   8. unpow-prod-down N/A

      \[\leadsto \frac{\frac{{d}^{2} \cdot {c0}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   10. associate-/l* N/A

       \[\leadsto \frac{{c0}^{2} \cdot \frac{{d}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{{c0}^{2} \cdot \frac{{d}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   12. unpow2 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{{d}^{2}}{h \cdot {w}^{2}}}{D \cdot D} \]

   14. associate-/r* N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{{d}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   15. lower-/.f64 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{{d}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   16. lower-/.f64 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{{d}^{2}}{h}}{{w}^{2}}}{D \cdot D} \]

   17. pow2 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{d \cdot d}{h}}{{w}^{2}}}{D \cdot D} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{d \cdot d}{h}}{{w}^{2}}}{D \cdot D} \]

   19. pow2 N/A

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{d \cdot d}{h}}{w \cdot w}}{D \cdot D} \]

   20. lift-*.f64 24.3

       \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{d \cdot d}{h}}{w \cdot w}}{D \cdot D} \]

10. Applied rewrites 24.3%

    \[\leadsto \frac{\left(c0 \cdot c0\right) \cdot \frac{\frac{d \cdot d}{h}}{w \cdot w}}{D \cdot D} \]
11. Add Preprocessing

Alternative 14: 0.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ c0 (+ w w)) (* (sqrt -1.0) M)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (sqrt(-1.0) * M);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (w + w)) * (sqrt((-1.0d0)) * m)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (w + w)) * (Math.sqrt(-1.0) * M);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (w + w)) * (math.sqrt(-1.0) * M)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(w + w)) * Float64(sqrt(-1.0) * M))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (w + w)) * (sqrt(-1.0) * M);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around 0

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot \color{blue}{M}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot \color{blue}{M}\right) \]

   3. lower-sqrt.f64 0.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\sqrt{-1} \cdot M\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{c0}{\color{blue}{2 \cdot w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   2. count-2-rev N/A

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

   3. lower-+.f64 0.0

      \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

7. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{\color{blue}{w + w}} \cdot \left(\sqrt{-1} \cdot M\right) \]

8. Final simplification 0.0%

   \[\leadsto \frac{c0}{w + w} \cdot \left(\sqrt{-1} \cdot M\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025083 
(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))))))