Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.3% → 50.9%

Time: 10.2s

Alternatives: 11

Speedup: 3.3×

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: 25.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	return (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
}

Fortran

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: 50.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(\frac{h \cdot w}{c0} \cdot \frac{\frac{{\left(M \cdot D\right)}^{2}}{d}}{d}\right) \cdot -0.5\right)}{w \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (/ (* 2.0 (* (* d c0) d)) (* (* (* h w) D) D)))
     (/
      (* c0 (* (* (/ (* h w) c0) (/ (/ (pow (* M D) 2.0) d) d)) -0.5))
      (* w 2.0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * ((((h * w) / c0) * ((pow((M * D), 2.0) / d) / d)) * -0.5)) / (w * 2.0);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * ((((h * w) / c0) * ((Math.pow((M * D), 2.0) / d) / d)) * -0.5)) / (w * 2.0);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D))
	else:
		tmp = (c0 * ((((h * w) / c0) * ((math.pow((M * D), 2.0) / d) / d)) * -0.5)) / (w * 2.0)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(Float64(d * c0) * d)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(c0 * Float64(Float64(Float64(Float64(h * w) / c0) * Float64(Float64((Float64(M * D) ^ 2.0) / d) / d)) * -0.5)) / Float64(w * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	else
		tmp = (c0 * ((((h * w) / c0) * ((((M * D) ^ 2.0) / d) / d)) * -0.5)) / (w * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[(N[(N[(N[(h * w), $MachinePrecision] / c0), $MachinePrecision] * N[(N[(N[Power[N[(M * D), $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.1

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

   7. Applied rewrites 76.1%

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

   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 h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 30.4

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

      13. lift-/.f64 N/A

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

   8. Applied rewrites 28.1%

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

   10. Applied rewrites 34.9%

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

Alternative 2: 43.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(-0.5 \cdot \left(\left(D \cdot D\right) \cdot \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot w}{\left(d \cdot d\right) \cdot c0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))) INFINITY)
     (* t_0 (/ (* 2.0 (* (* d c0) d)) (* (* (* h w) D) D)))
     (* t_0 (* -0.5 (* (* D D) (/ (* (* (* M M) h) w) (* (* d d) c0))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	} else {
		tmp = t_0 * (-0.5 * ((D * D) * ((((M * M) * h) * w) / ((d * d) * c0))));
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	} else {
		tmp = t_0 * (-0.5 * ((D * D) * ((((M * M) * h) * w) / ((d * d) * c0))));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))) <= math.inf:
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D))
	else:
		tmp = t_0 * (-0.5 * ((D * D) * ((((M * M) * h) * w) / ((d * d) * c0))))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(Float64(d * c0) * d)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(t_0 * Float64(-0.5 * Float64(Float64(D * D) * Float64(Float64(Float64(Float64(M * M) * h) * w) / Float64(Float64(d * d) * c0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * ((d * c0) * d)) / (((h * w) * D) * D));
	else
		tmp = t_0 * (-0.5 * ((D * D) * ((((M * M) * h) * w) / ((d * d) * c0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(-0.5 * N[(N[(D * D), $MachinePrecision] * N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.1

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

   7. Applied rewrites 76.1%

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

   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 h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 30.4

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

      13. lift-/.f64 N/A

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

   8. Applied rewrites 28.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. unpow-prod-down N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 N/A

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

   10. Applied rewrites 30.7%

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

Alternative 3: 44.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $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 73.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.1

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

   7. Applied rewrites 76.1%

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

   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 c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 16.8

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

   5. Applied rewrites 16.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

   7. Applied rewrites 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 26.6

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

   9. Applied rewrites 26.6%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 26.6

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

   11. Applied rewrites 26.6%

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

Alternative 4: 43.1% accurate, 0.7× speedup

Math

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

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)) (/ (* (* (* d c0) d) 2.0) (* (* (* D D) h) w)))
     (/ (/ (* (* d c0) (* d c0)) (* (* D w) (* D w))) h))))

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)) * ((((d * c0) * d) * 2.0) / (((D * D) * h) * w));
	} else {
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((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)) * ((((d * c0) * d) * 2.0) / (((D * D) * h) * w));
	} else {
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	}
	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)) * ((((d * c0) * d) * 2.0) / (((D * D) * h) * w))
	else:
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h
	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(Float64(Float64(d * c0) * d) * 2.0) / Float64(Float64(Float64(D * D) * h) * w)));
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(D * w) * Float64(D * w))) / h);
	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)) * ((((d * c0) * d) * 2.0) / (((D * D) * h) * w));
	else
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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[(N[(N[(d * c0), $MachinePrecision] * d), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $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 73.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 72.8

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

   5. Applied rewrites 72.8%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]
   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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 72.8

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. pow2 N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 71.6

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

   7. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 74.9

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

   9. Applied rewrites 74.9%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 74.9

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

   11. Applied rewrites 74.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 16.8

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

   5. Applied rewrites 16.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

   7. Applied rewrites 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 26.6

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

   9. Applied rewrites 26.6%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 26.6

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

   11. Applied rewrites 26.6%

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

Alternative 5: 43.7% accurate, 0.7× speedup

Math

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

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 d) c0)) (* (* (* h w) D) D)))
     (/ (/ (* (* d c0) (* d c0)) (* (* D w) (* D w))) h))))

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 * d) * c0)) / (((h * w) * D) * D));
	} else {
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((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 * d) * c0)) / (((h * w) * D) * D));
	} else {
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	}
	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 * d) * c0)) / (((h * w) * D) * D))
	else:
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h
	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 * d) * c0)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(D * w) * Float64(D * w))) / h);
	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 * d) * c0)) / (((h * w) * D) * D));
	else
		tmp = (((d * c0) * (d * c0)) / ((D * w) * (D * w))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(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 * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(D * w), $MachinePrecision] * N[(D * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $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 73.2%

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

      1. lift-*.f64 N/A

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

      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 72.8

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

   7. Applied rewrites 72.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 16.8

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

   5. Applied rewrites 16.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

   7. Applied rewrites 26.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 26.6

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

   9. Applied rewrites 26.6%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 26.6

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

   11. Applied rewrites 26.6%

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

Alternative 6: 34.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(w * w) * h) * Float64(D * D))
	t_1 = 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_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(c0 * Float64(Float64(Float64(d * d) * c0) / t_0));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / t_0)));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-/.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 64.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 16.8

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

   5. Applied rewrites 16.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 11.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 18.1

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

   9. Applied rewrites 18.1%

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

Alternative 7: 34.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot w\right) \cdot h\\ \mathbf{if}\;d \leq 10^{-117}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{t\_0 \cdot \left(D \cdot D\right)}\right)\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{+145}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot D}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* w w) h)))
   (if (<= d 1e-117)
     (* (* c0 c0) (* d (/ d (* t_0 (* D D)))))
     (if (<= d 1.65e+145)
       (* (* c0 c0) (/ (* d d) (* (* (* (* h w) D) D) w)))
       (/ (/ (* (* d c0) (* d c0)) (* D D)) t_0)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * w) * h;
	double tmp;
	if (d <= 1e-117) {
		tmp = (c0 * c0) * (d * (d / (t_0 * (D * D))));
	} else if (d <= 1.65e+145) {
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w));
	} else {
		tmp = (((d * c0) * (d * c0)) / (D * D)) / t_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) :: t_0
    real(8) :: tmp
    t_0 = (w * w) * h
    if (d_1 <= 1d-117) then
        tmp = (c0 * c0) * (d_1 * (d_1 / (t_0 * (d * d))))
    else if (d_1 <= 1.65d+145) then
        tmp = (c0 * c0) * ((d_1 * d_1) / ((((h * w) * d) * d) * w))
    else
        tmp = (((d_1 * c0) * (d_1 * c0)) / (d * d)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * w) * h;
	double tmp;
	if (d <= 1e-117) {
		tmp = (c0 * c0) * (d * (d / (t_0 * (D * D))));
	} else if (d <= 1.65e+145) {
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w));
	} else {
		tmp = (((d * c0) * (d * c0)) / (D * D)) / t_0;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (w * w) * h
	tmp = 0
	if d <= 1e-117:
		tmp = (c0 * c0) * (d * (d / (t_0 * (D * D))))
	elif d <= 1.65e+145:
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w))
	else:
		tmp = (((d * c0) * (d * c0)) / (D * D)) / t_0
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(w * w) * h)
	tmp = 0.0
	if (d <= 1e-117)
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / Float64(t_0 * Float64(D * D)))));
	elseif (d <= 1.65e+145)
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(Float64(Float64(h * w) * D) * D) * w)));
	else
		tmp = Float64(Float64(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(D * D)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (w * w) * h;
	tmp = 0.0;
	if (d <= 1e-117)
		tmp = (c0 * c0) * (d * (d / (t_0 * (D * D))));
	elseif (d <= 1.65e+145)
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w));
	else
		tmp = (((d * c0) * (d * c0)) / (D * D)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision]}, If[LessEqual[d, 1e-117], N[(N[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(t$95$0 * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.65e+145], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < 1.00000000000000003e-117

   1. Initial program 24.5%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 34.1

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

   5. Applied rewrites 34.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 28.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 34.8

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

   9. Applied rewrites 34.8%

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

   if 1.00000000000000003e-117 < d < 1.65000000000000013e145

   1. Initial program 30.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 29.3

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

   5. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 40.0

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

   9. Applied rewrites 40.0%

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

   if 1.65000000000000013e145 < d

   1. Initial program 20.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 33.1

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

   7. Applied rewrites 33.1%

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

Alternative 8: 34.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 10^{-117} \lor \neg \left(d \leq 4 \cdot 10^{+145}\right):\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (or (<= d 1e-117) (not (<= d 4e+145)))
   (* (* c0 c0) (* d (/ d (* (* (* w w) h) (* D D)))))
   (* (* c0 c0) (/ (* d d) (* (* (* (* h w) D) D) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((d <= 1e-117) || !(d <= 4e+145)) {
		tmp = (c0 * c0) * (d * (d / (((w * w) * h) * (D * D))));
	} else {
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w));
	}
	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 <= 1d-117) .or. (.not. (d_1 <= 4d+145))) then
        tmp = (c0 * c0) * (d_1 * (d_1 / (((w * w) * h) * (d * d))))
    else
        tmp = (c0 * c0) * ((d_1 * d_1) / ((((h * w) * d) * d) * w))
    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 <= 1e-117) || !(d <= 4e+145)) {
		tmp = (c0 * c0) * (d * (d / (((w * w) * h) * (D * D))));
	} else {
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w));
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (d <= 1e-117) or not (d <= 4e+145):
		tmp = (c0 * c0) * (d * (d / (((w * w) * h) * (D * D))))
	else:
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w))
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if ((d <= 1e-117) || !(d <= 4e+145))
		tmp = Float64(Float64(c0 * c0) * Float64(d * Float64(d / Float64(Float64(Float64(w * w) * h) * Float64(D * D)))));
	else
		tmp = Float64(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(Float64(Float64(h * w) * D) * D) * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((d <= 1e-117) || ~((d <= 4e+145)))
		tmp = (c0 * c0) * (d * (d / (((w * w) * h) * (D * D))));
	else
		tmp = (c0 * c0) * ((d * d) / ((((h * w) * D) * D) * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[Or[LessEqual[d, 1e-117], N[Not[LessEqual[d, 4e+145]], $MachinePrecision]], N[(N[(c0 * c0), $MachinePrecision] * N[(d * N[(d / N[(N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.00000000000000003e-117 or 4e145 < d

   1. Initial program 23.3%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.8

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

   5. Applied rewrites 33.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 26.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 33.4

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

   9. Applied rewrites 33.4%

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

   if 1.00000000000000003e-117 < d < 4e145

   1. Initial program 30.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 29.3

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

   5. Applied rewrites 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 29.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 40.0

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

   9. Applied rewrites 40.0%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 10^{-117} \lor \neg \left(d \leq 4 \cdot 10^{+145}\right):\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \left(d \cdot \frac{d}{\left(\left(w \cdot w\right) \cdot h\right) \cdot \left(D \cdot D\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(\left(h \cdot w\right) \cdot D\right) \cdot D\right) \cdot w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

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)) / ((d * w) * (d * w))) / h
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in c0 around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 32.7

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

5. Applied rewrites 32.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow-prod-down N/A

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

   11. pow2 N/A

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

   12. associate-/r* N/A

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

7. Applied rewrites 40.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 40.1

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

9. Applied rewrites 40.1%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 40.1

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

11. Applied rewrites 40.1%

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

Alternative 10: 29.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return c0 * (((d * d) * c0) / (((w * w) * h) * (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 * (((d_1 * d_1) * c0) / (((w * w) * h) * (d * d)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in c0 around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 32.7

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

5. Applied rewrites 32.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-prod-down N/A

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

   7. pow2 N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 N/A

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

   20. associate-*r* N/A

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

   21. lower-*.f64 N/A

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

7. Applied rewrites 27.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-/.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

9. Applied rewrites 29.6%

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

Alternative 11: 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 25.2%

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

Reproduce

herbie shell --seed 2025072 
(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))))))