Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 61.3%

Time: 11.9s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: t_0
    t_0 = (c0 * (d_1 * d_1)) / ((w * h) * (d * d))
    code = (c0 / (2.0d0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (m * m))))
end function

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	return (c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = (c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
end

Wolfram

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

Initial Program: 24.3% 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: 61.3% accurate, N/A× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (pow (* D M) 1.0))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_0 (/ (* 2.0 (* d (* d c0))) (* (* (* h w) D) D)))
     (* (/ (* (* t_1 t_1) h) (* d d)) 0.25))))

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 = pow((D * M), 1.0);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	} else {
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = Math.pow((D * M), 1.0);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	} else {
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = math.pow((D * M), 1.0)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_0 * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D))
	else:
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(D * M) ^ 1.0
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(d * Float64(d * c0))) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 * t_1) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (D * M) ^ 1.0;
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	else
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25;
	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[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$2 = 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$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 61.3% accurate, N/A× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (pow (* D M) 1.0))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_0 (/ (* 2.0 (* d (* d c0))) (* (* h (* w D)) D)))
     (* (/ (* (* t_1 t_1) h) (* d d)) 0.25))))

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 = pow((D * M), 1.0);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * (d * (d * c0))) / ((h * (w * D)) * D));
	} else {
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = Math.pow((D * M), 1.0);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * (d * (d * c0))) / ((h * (w * D)) * D));
	} else {
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = math.pow((D * M), 1.0)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_0 * ((2.0 * (d * (d * c0))) / ((h * (w * D)) * D))
	else:
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(D * M) ^ 1.0
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(Float64(2.0 * Float64(d * Float64(d * c0))) / Float64(Float64(h * Float64(w * D)) * D)));
	else
		tmp = Float64(Float64(Float64(Float64(t_1 * t_1) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (D * M) ^ 1.0;
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_0 * ((2.0 * (d * (d * c0))) / ((h * (w * D)) * D));
	else
		tmp = (((t_1 * t_1) * h) / (d * d)) * 0.25;
	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[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$2 = 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$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(h * N[(w * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      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. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.5

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

   9. Applied rewrites 75.5%

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

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 61.0% accurate, N/A× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (pow (* D M) 1.0)))
   (if (<= (* t_1 (+ t_0 (sqrt (- (* t_0 t_0) (* M M))))) INFINITY)
     (* t_1 (* (/ 2.0 (* (* h w) D)) (/ (* (* d d) c0) D)))
     (* (/ (* (* t_2 t_2) h) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = pow((D * M), 1.0);
	double tmp;
	if ((t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D));
	} else {
		tmp = (((t_2 * t_2) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = Math.pow((D * M), 1.0);
	double tmp;
	if ((t_1 * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D));
	} else {
		tmp = (((t_2 * t_2) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_1 = c0 / (2.0 * w)
	t_2 = math.pow((D * M), 1.0)
	tmp = 0
	if (t_1 * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = t_1 * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D))
	else:
		tmp = (((t_2 * t_2) * h) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(D * M) ^ 1.0
	tmp = 0.0
	if (Float64(t_1 * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(2.0 / Float64(Float64(h * w) * D)) * Float64(Float64(Float64(d * d) * c0) / D)));
	else
		tmp = Float64(Float64(Float64(Float64(t_2 * t_2) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_1 = c0 / (2.0 * w);
	t_2 = (D * M) ^ 1.0;
	tmp = 0.0;
	if ((t_1 * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = t_1 * ((2.0 / ((h * w) * D)) * (((d * d) * c0) / D));
	else
		tmp = (((t_2 * t_2) * h) / (d * d)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, If[LessEqual[N[(t$95$1 * 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[(t$95$1 * N[(N[(2.0 / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 75.4

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

   7. Applied rewrites 75.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(D \cdot M\right)}^{1}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\left(-1 \cdot c0\right) \cdot \mathsf{fma}\left(-2, \frac{\frac{d}{D \cdot D}}{h} \cdot \frac{d}{w}, \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w\right) \cdot \left(D \cdot D\right)}{{\left(d \cdot c0\right)}^{2}} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_0\right) \cdot h}{d \cdot d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* D M) 1.0))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (*
      t_1
      (*
       (* -1.0 c0)
       (fma
        -2.0
        (* (/ (/ d (* D D)) h) (/ d w))
        (* (/ (* (* (* (* M M) h) w) (* D D)) (pow (* d c0) 2.0)) 0.5))))
     (* (/ (* (* t_0 t_0) h) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((D * M), 1.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((-1.0 * c0) * fma(-2.0, (((d / (D * D)) / h) * (d / w)), ((((((M * M) * h) * w) * (D * D)) / pow((d * c0), 2.0)) * 0.5)));
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * M) ^ 1.0
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(-1.0 * c0) * fma(-2.0, Float64(Float64(Float64(d / Float64(D * D)) / h) * Float64(d / w)), Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * w) * Float64(D * D)) / (Float64(d * c0) ^ 2.0)) * 0.5))));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(-1.0 * c0), $MachinePrecision] * N[(-2.0 * N[(N[(N[(d / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in 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 58.0%

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

   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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 10.2

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

   8. Applied rewrites 10.2%

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

   9. Taylor expanded in c0 around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 69.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-/r* N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 71.2

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

   13. Applied rewrites 71.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(D \cdot M\right)}^{1}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\left(-1 \cdot c0\right) \cdot \mathsf{fma}\left(-2, \frac{d}{\left(D \cdot D\right) \cdot h} \cdot \frac{d}{w}, \frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot w\right) \cdot \left(D \cdot D\right)}{{\left(d \cdot c0\right)}^{2}} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_0\right) \cdot h}{d \cdot d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* D M) 1.0))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (*
      t_1
      (*
       (* -1.0 c0)
       (fma
        -2.0
        (* (/ d (* (* D D) h)) (/ d w))
        (* (/ (* (* (* (* M M) h) w) (* D D)) (pow (* d c0) 2.0)) 0.5))))
     (* (/ (* (* t_0 t_0) h) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((D * M), 1.0);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((-1.0 * c0) * fma(-2.0, ((d / ((D * D) * h)) * (d / w)), ((((((M * M) * h) * w) * (D * D)) / pow((d * c0), 2.0)) * 0.5)));
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * M) ^ 1.0
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(-1.0 * c0) * fma(-2.0, Float64(Float64(d / Float64(Float64(D * D) * h)) * Float64(d / w)), Float64(Float64(Float64(Float64(Float64(Float64(M * M) * h) * w) * Float64(D * D)) / (Float64(d * c0) ^ 2.0)) * 0.5))));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(-1.0 * c0), $MachinePrecision] * N[(-2.0 * N[(N[(d / N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d / w), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in 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 58.0%

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

   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

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 10.2

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

   8. Applied rewrites 10.2%

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

   9. Taylor expanded in c0 around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 69.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* D M) 1.0)) (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) (* c0 d)) (* h (* w w))) (* D D))
     (* (/ (* (* t_0 t_0) h) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((D * M), 1.0);
	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) * (c0 * d)) / (h * (w * w))) / (D * D);
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((D * M), 1.0);
	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) * (c0 * d)) / (h * (w * w))) / (D * D);
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((D * M), 1.0)
	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) * (c0 * d)) / (h * (w * w))) / (D * D)
	else:
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * M) ^ 1.0
	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(Float64(Float64(Float64(c0 * d) * Float64(c0 * d)) / Float64(h * Float64(w * w))) / Float64(D * D));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * M) ^ 1.0;
	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) * (c0 * d)) / (h * (w * w))) / (D * D);
	else
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 1.0], $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[(N[(N[(N[(c0 * d), $MachinePrecision] * N[(c0 * d), $MachinePrecision]), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

   3. Taylor expanded in D around 0

      \[\leadsto \color{blue}{\frac{{D}^{4} \cdot \left(\frac{-1}{4} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}} + \frac{-1}{16} \cdot \frac{{D}^{4} \cdot \left({M}^{4} \cdot \left({h}^{3} \cdot {w}^{2}\right)\right)}{{c0}^{2} \cdot {d}^{6}}\right) + \frac{{c0}^{2} \cdot {d}^{2}}{h \cdot {w}^{2}}}{{D}^{2}}} \]

   5. Applied rewrites 25.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{D}^{4}}{c0 \cdot c0} \cdot \frac{\left(\left(\left(h \cdot h\right) \cdot h\right) \cdot \left(w \cdot w\right)\right) \cdot {M}^{4}}{{d}^{6}}, -0.0625, \frac{-0.25 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}\right), {D}^{4}, \frac{{\left(c0 \cdot d\right)}^{1} \cdot {\left(c0 \cdot d\right)}^{1}}{\left(w \cdot w\right) \cdot h}\right)}{D \cdot D}} \]

   6. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow-prod-down N/A

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

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. unpow1 N/A

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

      6. unpow1 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 67.4

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

   8. Applied rewrites 67.4%

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

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.5% accurate, N/A× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* D M) 1.0)) (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 (* c0 (* (/ d (* (* D D) h)) (/ d (* w w)))))
     (* (/ (* (* t_0 t_0) h) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((D * M), 1.0);
	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 * (c0 * ((d / ((D * D) * h)) * (d / (w * w))));
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = Math.pow((D * M), 1.0);
	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 * (c0 * ((d / ((D * D) * h)) * (d / (w * w))));
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((D * M), 1.0)
	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 * (c0 * ((d / ((D * D) * h)) * (d / (w * w))))
	else:
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * M) ^ 1.0
	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(c0 * Float64(Float64(d / Float64(Float64(D * D) * h)) * Float64(d / Float64(w * w)))));
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * M) ^ 1.0;
	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 * (c0 * ((d / ((D * D) * h)) * (d / (w * w))));
	else
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 1.0], $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[(c0 * N[(N[(d / N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(d / N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      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

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. associate-*r* N/A

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

   7. Applied rewrites 63.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 15.6

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

   5. Applied rewrites 15.6%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 42.8

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

   11. Applied rewrites 42.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 54.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 54.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 41.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(D \cdot M\right)}^{1}\\ \mathbf{if}\;d \leq 8.8 \cdot 10^{-95}:\\ \;\;\;\;\left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{D \cdot D}{d}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot t\_0\right) \cdot h}{d \cdot d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (pow (* D M) 1.0)))
   (if (<= d 8.8e-95)
     (* (* (/ (* (* M h) M) d) (/ (* D D) d)) 0.25)
     (* (/ (* (* t_0 t_0) h) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = pow((D * M), 1.0);
	double tmp;
	if (d <= 8.8e-95) {
		tmp = ((((M * h) * M) / d) * ((D * D) / d)) * 0.25;
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	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 = (d * m) ** 1.0d0
    if (d_1 <= 8.8d-95) then
        tmp = ((((m * h) * m) / d_1) * ((d * d) / d_1)) * 0.25d0
    else
        tmp = (((t_0 * t_0) * h) / (d_1 * d_1)) * 0.25d0
    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 = Math.pow((D * M), 1.0);
	double tmp;
	if (d <= 8.8e-95) {
		tmp = ((((M * h) * M) / d) * ((D * D) / d)) * 0.25;
	} else {
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = math.pow((D * M), 1.0)
	tmp = 0
	if d <= 8.8e-95:
		tmp = ((((M * h) * M) / d) * ((D * D) / d)) * 0.25
	else:
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(D * M) ^ 1.0
	tmp = 0.0
	if (d <= 8.8e-95)
		tmp = Float64(Float64(Float64(Float64(Float64(M * h) * M) / d) * Float64(Float64(D * D) / d)) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) * h) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (D * M) ^ 1.0;
	tmp = 0.0;
	if (d <= 8.8e-95)
		tmp = ((((M * h) * M) / d) * ((D * D) / d)) * 0.25;
	else
		tmp = (((t_0 * t_0) * h) / (d * d)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[Power[N[(D * M), $MachinePrecision], 1.0], $MachinePrecision]}, If[LessEqual[d, 8.8e-95], N[(N[(N[(N[(N[(M * h), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * h), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 8.7999999999999995e-95

   1. Initial program 20.3%

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

   3. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
   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 30.0

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

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 33.8

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

   11. Applied rewrites 33.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 39.5

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

   13. Applied rewrites 39.5%

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

   if 8.7999999999999995e-95 < d

   1. Initial program 23.1%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 4.6%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 35.7

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

   11. Applied rewrites 35.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. pow-prod-down N/A

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

       11. metadata-eval N/A

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

       12. unpow-prod-up N/A

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

       13. lower-*.f64 N/A

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

       14. lower-pow.f64 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-pow.f64 N/A

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

       17. lower-*.f64 43.4

           \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]

   13. Applied rewrites 43.4%

       \[\leadsto \frac{\left({\left(D \cdot M\right)}^{1} \cdot {\left(D \cdot M\right)}^{1}\right) \cdot h}{d \cdot d} \cdot 0.25 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 40.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(M \cdot h\right) \cdot M\\ \mathbf{if}\;d \leq 1.2 \cdot 10^{-137}:\\ \;\;\;\;\left(\frac{t\_0}{d} \cdot \frac{D \cdot D}{d}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 \cdot D\right) \cdot D}{d \cdot d} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* M h) M)))
   (if (<= d 1.2e-137)
     (* (* (/ t_0 d) (/ (* D D) d)) 0.25)
     (* (/ (* (* t_0 D) D) (* d d)) 0.25))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (M * h) * M;
	double tmp;
	if (d <= 1.2e-137) {
		tmp = ((t_0 / d) * ((D * D) / d)) * 0.25;
	} else {
		tmp = (((t_0 * D) * D) / (d * d)) * 0.25;
	}
	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 = (m * h) * m
    if (d_1 <= 1.2d-137) then
        tmp = ((t_0 / d_1) * ((d * d) / d_1)) * 0.25d0
    else
        tmp = (((t_0 * d) * d) / (d_1 * d_1)) * 0.25d0
    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 = (M * h) * M;
	double tmp;
	if (d <= 1.2e-137) {
		tmp = ((t_0 / d) * ((D * D) / d)) * 0.25;
	} else {
		tmp = (((t_0 * D) * D) / (d * d)) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (M * h) * M
	tmp = 0
	if d <= 1.2e-137:
		tmp = ((t_0 / d) * ((D * D) / d)) * 0.25
	else:
		tmp = (((t_0 * D) * D) / (d * d)) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(M * h) * M)
	tmp = 0.0
	if (d <= 1.2e-137)
		tmp = Float64(Float64(Float64(t_0 / d) * Float64(Float64(D * D) / d)) * 0.25);
	else
		tmp = Float64(Float64(Float64(Float64(t_0 * D) * D) / Float64(d * d)) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (M * h) * M;
	tmp = 0.0;
	if (d <= 1.2e-137)
		tmp = ((t_0 / d) * ((D * D) / d)) * 0.25;
	else
		tmp = (((t_0 * D) * D) / (d * d)) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(M * h), $MachinePrecision] * M), $MachinePrecision]}, If[LessEqual[d, 1.2e-137], N[(N[(N[(t$95$0 / d), $MachinePrecision] * N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(N[(t$95$0 * D), $MachinePrecision] * D), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 1.2e-137

   1. Initial program 19.0%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 33.7

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

   11. Applied rewrites 33.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. times-frac N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 38.9

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

   13. Applied rewrites 38.9%

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

   if 1.2e-137 < d

   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

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 37.9

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 4.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

   9. Taylor expanded in c0 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 35.8

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

   11. Applied rewrites 35.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 38.6

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

   13. Applied rewrites 38.6%

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

Alternative 10: 39.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in c0 around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
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 33.1

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

5. Applied rewrites 33.1%

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

6. Taylor expanded in c0 around -inf

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

8. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

9. Taylor expanded in c0 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 N/A

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

    11. pow2 N/A

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

    12. lift-*.f64 34.6

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

11. Applied rewrites 34.6%

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

    1. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    7. pow2 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{d \cdot d} \cdot \frac{1}{4} \]

    8. times-frac N/A

       \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    10. lower-/.f64 N/A

        \[\leadsto \left(\frac{\left(M \cdot M\right) \cdot h}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    11. associate-*r* N/A

        \[\leadsto \left(\frac{M \cdot \left(M \cdot h\right)}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    12. *-commutative N/A

        \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    14. lift-*.f64 N/A

        \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    15. lower-/.f64 N/A

        \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{{D}^{2}}{d}\right) \cdot \frac{1}{4} \]

    16. pow2 N/A

        \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{D \cdot D}{d}\right) \cdot \frac{1}{4} \]

    17. lift-*.f64 39.7

        \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{D \cdot D}{d}\right) \cdot 0.25 \]

13. Applied rewrites 39.7%

    \[\leadsto \left(\frac{\left(M \cdot h\right) \cdot M}{d} \cdot \frac{D \cdot D}{d}\right) \cdot 0.25 \]
14. Add Preprocessing

Alternative 11: 37.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(\left(D \cdot D\right) \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.25 \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (* (* D D) (* (/ (* M M) d) (/ h d))) 0.25))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((D * D) * (((M * M) / d) * (h / d))) * 0.25;
}

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 * d) * (((m * m) / d_1) * (h / d_1))) * 0.25d0
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((D * D) * (((M * M) / d) * (h / d))) * 0.25;
}

Python

def code(c0, w, h, D, d, M):
	return ((D * D) * (((M * M) / d) * (h / d))) * 0.25

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(D * D) * Float64(Float64(Float64(M * M) / d) * Float64(h / d))) * 0.25)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((D * D) * (((M * M) / d) * (h / d))) * 0.25;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]

Derivation

1. Initial program 21.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]
2. Add Preprocessing

3. Taylor expanded in c0 around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}\right)} \]
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 33.1

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} \]

5. Applied rewrites 33.1%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}} \]

6. Taylor expanded in c0 around -inf

   \[\leadsto \color{blue}{{c0}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {c0}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right)} \]

   2. pow2 N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot \frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}} + \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{-1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)} + \frac{{d}^{2}}{{D}^{2} \cdot \left(h \cdot w\right)}}{w}}, \frac{1}{4} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{c0}^{2} \cdot {d}^{2}}\right) \]

8. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, 0.25 \cdot \left(\frac{D \cdot D}{c0 \cdot c0} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right)\right)} \]

9. Taylor expanded in c0 around 0

   \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    3. lower-/.f64 N/A

       \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    4. *-commutative N/A

       \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

    6. pow2 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

    8. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

    9. pow2 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    10. lift-*.f64 N/A

        \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    11. pow2 N/A

        \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    12. lift-*.f64 34.6

        \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.25 \]

11. Applied rewrites 34.6%

    \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \color{blue}{0.25} \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{4} \]

    3. pow2 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    7. pow2 N/A

       \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    8. lift-*.f64 N/A

       \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    9. pow2 N/A

       \[\leadsto \frac{\left({M}^{2} \cdot h\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{4} \]

    10. *-commutative N/A

        \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}} \cdot \frac{1}{4} \]

    11. associate-/l* N/A

        \[\leadsto \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{{d}^{2}}\right) \cdot \frac{1}{4} \]

    12. pow2 N/A

        \[\leadsto \left({D}^{2} \cdot \frac{{M}^{2} \cdot h}{d \cdot d}\right) \cdot \frac{1}{4} \]

    13. pow2 N/A

        \[\leadsto \left({D}^{2} \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot \frac{1}{4} \]

    14. associate-*r* N/A

        \[\leadsto \left({D}^{2} \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot d}\right) \cdot \frac{1}{4} \]

    15. lower-*.f64 N/A

        \[\leadsto \left({D}^{2} \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot d}\right) \cdot \frac{1}{4} \]

    16. pow2 N/A

        \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot d}\right) \cdot \frac{1}{4} \]

    17. lift-*.f64 N/A

        \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{M \cdot \left(M \cdot h\right)}{d \cdot d}\right) \cdot \frac{1}{4} \]

    18. associate-*r* N/A

        \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{d \cdot d}\right) \cdot \frac{1}{4} \]

    19. pow2 N/A

        \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{{M}^{2} \cdot h}{d \cdot d}\right) \cdot \frac{1}{4} \]

    20. times-frac N/A

        \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)\right) \cdot \frac{1}{4} \]

    21. lower-*.f64 N/A

        \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\frac{{M}^{2}}{d} \cdot \frac{h}{d}\right)\right) \cdot \frac{1}{4} \]

13. Applied rewrites 36.6%

    \[\leadsto \left(\left(D \cdot D\right) \cdot \left(\frac{M \cdot M}{d} \cdot \frac{h}{d}\right)\right) \cdot 0.25 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))))