Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 46.2%

Time: 9.5s

Alternatives: 17

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% 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: 46.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(w \cdot h\right) \cdot \left(D \cdot D\right)\\ t_1 := \frac{\left(c0 \cdot d\right) \cdot d}{t\_0}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d \cdot d\right)}{t\_0}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (* w h) (* D D)))
        (t_1 (/ (* (* c0 d) d) t_0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d d)) t_0)))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M M))))) INFINITY)
     (* t_2 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))
     (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * h) * (D * D);
	double t_1 = ((c0 * d) * d) / t_0;
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / t_0;
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (w * h) * (D * D);
	double t_1 = ((c0 * d) * d) / t_0;
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d * d)) / t_0;
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = (w * h) * (D * D)
	t_1 = ((c0 * d) * d) / t_0
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d * d)) / t_0
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M * M))))) <= math.inf:
		tmp = t_2 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(w * h) * Float64(D * D))
	t_1 = Float64(Float64(Float64(c0 * d) * d) / t_0)
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d * d)) / t_0)
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_2 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (w * h) * (D * D);
	t_1 = ((c0 * d) * d) / t_0;
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d * d)) / t_0;
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M * M))))) <= Inf)
		tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c0 * d), $MachinePrecision] * d), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 24.8

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

   3. Applied rewrites 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 25.0

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

   5. Applied rewrites 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 27.5

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

   7. Applied rewrites 27.5%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 2: 45.8% accurate, 0.5× speedup

Math

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

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_0 (sqrt (- (* t_0 t_0) (* M M)))))))
   (if (<= t_1 INFINITY) t_1 (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w)))))

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)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	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)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	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_0 + math.sqrt(((t_0 * t_0) - (M * M))))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	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(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M)))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	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_0 + sqrt(((t_0 * t_0) - (M * M))));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	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[(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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 3: 45.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 * ((d / ((w * h) * D)) * (d / D));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * (t_0 + Math.sqrt((Math.pow(t_0, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 * ((d / ((w * h) * D)) * (d / D))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * (t_0 + math.sqrt((math.pow(t_0, 2.0) - (M * M))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 * ((d / ((w * h) * D)) * (d / D));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * (t_0 + sqrt(((t_0 ^ 2.0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   3. Applied rewrites 28.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 35.5

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

   7. Applied rewrites 35.5%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 4: 45.4% accurate, 0.5× speedup

Math

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

FPCore

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

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 * (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 / (w + w)) * (t_0 + sqrt((pow(t_0, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	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 * (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 / (w + w)) * (t_0 + Math.sqrt((Math.pow(t_0, 2.0) - (M * M))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(w * h) * D) * D)))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(t_0 + sqrt(Float64((t_0 ^ 2.0) - Float64(M * M)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	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 * (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 / (w + w)) * (t_0 + sqrt(((t_0 ^ 2.0) - (M * M))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   3. Applied rewrites 27.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-+.f64 28.1

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

   5. Applied rewrites 28.1%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 5: 44.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right)\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{-79}:\\ \;\;\;\;t\_1 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(h \cdot \sqrt{{D}^{4}}\right)} + t\_0\right)\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(\left(D \cdot D\right) \cdot \sqrt{h \cdot h}\right)} + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* (* (* D D) h) w)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M)))))))
   (if (<= t_3 -4e-79)
     (* t_1 (* (* d d) (+ (/ c0 (* w (* h (sqrt (pow D 4.0))))) t_0)))
     (if (<= t_3 INFINITY)
       (* t_1 (* (* d d) (+ (/ c0 (* w (* (* D D) (sqrt (* h h))))) t_0)))
       (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (((D * D) * h) * w);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e-79) {
		tmp = t_1 * ((d * d) * ((c0 / (w * (h * sqrt(pow(D, 4.0))))) + t_0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + t_0));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	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) * h) * w);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_3 = t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))));
	double tmp;
	if (t_3 <= -4e-79) {
		tmp = t_1 * ((d * d) * ((c0 / (w * (h * Math.sqrt(Math.pow(D, 4.0))))) + t_0));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((d * d) * ((c0 / (w * ((D * D) * Math.sqrt((h * h))))) + t_0));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (((D * D) * h) * w)
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_3 = t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))
	tmp = 0
	if t_3 <= -4e-79:
		tmp = t_1 * ((d * d) * ((c0 / (w * (h * math.sqrt(math.pow(D, 4.0))))) + t_0))
	elif t_3 <= math.inf:
		tmp = t_1 * ((d * d) * ((c0 / (w * ((D * D) * math.sqrt((h * h))))) + t_0))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(Float64(Float64(D * D) * h) * w))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M)))))
	tmp = 0.0
	if (t_3 <= -4e-79)
		tmp = Float64(t_1 * Float64(Float64(d * d) * Float64(Float64(c0 / Float64(w * Float64(h * sqrt((D ^ 4.0))))) + t_0)));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * Float64(Float64(d * d) * Float64(Float64(c0 / Float64(w * Float64(Float64(D * D) * sqrt(Float64(h * h))))) + t_0)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (((D * D) * h) * w);
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))));
	tmp = 0.0;
	if (t_3 <= -4e-79)
		tmp = t_1 * ((d * d) * ((c0 / (w * (h * sqrt((D ^ 4.0))))) + t_0));
	elseif (t_3 <= Inf)
		tmp = t_1 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + t_0));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $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]}, Block[{t$95$3 = 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]}, If[LessEqual[t$95$3, -4e-79], N[(t$95$1 * N[(N[(d * d), $MachinePrecision] * N[(N[(c0 / N[(w * N[(h * N[Sqrt[N[Power[D, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[(d * d), $MachinePrecision] * N[(N[(c0 / N[(w * N[(N[(D * D), $MachinePrecision] * N[Sqrt[N[(h * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))))) < -4e-79

   1. Initial program 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{\left(\frac{4}{2}\right)} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{2} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in h around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(h \cdot \sqrt{{D}^{4}}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(h \cdot \sqrt{{D}^{4}}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

       3. lower-pow.f64 32.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(h \cdot \sqrt{{D}^{4}}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

   13. Applied rewrites 32.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(h \cdot \sqrt{{D}^{4}}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

   if -4e-79 < (*.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 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{\left(\frac{4}{2}\right)} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{2} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in D around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 16.6

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

   13. Applied rewrites 16.6%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 6: 42.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   3. Applied rewrites 27.3%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 7: 41.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-79}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 + \sqrt{M \cdot \left(c0 \cdot \frac{d \cdot d}{\left(\left(w \cdot h\right) \cdot D\right) \cdot D} - M\right)}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(\left(D \cdot D\right) \cdot \sqrt{h \cdot h}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -4e-79)
     (* t_0 (+ t_1 (sqrt (* M (- (* c0 (/ (* d d) (* (* (* w h) D) D))) M)))))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (+
          (/ c0 (* w (* (* D D) (sqrt (* h h)))))
          (/ c0 (* (* (* D D) h) w)))))
       (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-79) {
		tmp = t_0 * (t_1 + sqrt((M * ((c0 * ((d * d) / (((w * h) * D) * D))) - M))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-79) {
		tmp = t_0 * (t_1 + Math.sqrt((M * ((c0 * ((d * d) / (((w * h) * D) * D))) - M))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * Math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -4e-79:
		tmp = t_0 * (t_1 + math.sqrt((M * ((c0 * ((d * d) / (((w * h) * D) * D))) - M))))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -4e-79)
		tmp = Float64(t_0 * Float64(t_1 + sqrt(Float64(M * Float64(Float64(c0 * Float64(Float64(d * d) / Float64(Float64(Float64(w * h) * D) * D))) - M)))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(c0 / Float64(w * Float64(Float64(D * D) * sqrt(Float64(h * h))))) + Float64(c0 / Float64(Float64(Float64(D * D) * h) * w)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -4e-79)
		tmp = t_0 * (t_1 + sqrt((M * ((c0 * ((d * d) / (((w * h) * D) * D))) - M))));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-79], N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(M * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(c0 / N[(w * N[(N[(D * D), $MachinePrecision] * N[Sqrt[N[(h * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -4e-79

   1. Initial program 25.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.9%

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

   4. Taylor expanded in c0 around 0

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

   6. Applied rewrites 15.2%

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

   if -4e-79 < (*.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 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{\left(\frac{4}{2}\right)} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{2} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in D around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 16.6

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

   13. Applied rewrites 16.6%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 8: 41.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-79}:\\ \;\;\;\;t\_0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot w\right) \cdot \left(h \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{D \cdot D}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(\left(D \cdot D\right) \cdot \sqrt{h \cdot h}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -4e-79)
     (*
      t_0
      (/
       (* (* d d) (+ (sqrt (/ (* c0 c0) (* (* h w) (* h w)))) (/ c0 (* h w))))
       (* D D)))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (+
          (/ c0 (* w (* (* D D) (sqrt (* h h)))))
          (/ c0 (* (* (* D D) h) w)))))
       (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-79) {
		tmp = t_0 * (((d * d) * (sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / (D * D));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-79) {
		tmp = t_0 * (((d * d) * (Math.sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / (D * D));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * Math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -4e-79:
		tmp = t_0 * (((d * d) * (math.sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / (D * D))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -4e-79)
		tmp = Float64(t_0 * Float64(Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(Float64(h * w) * Float64(h * w)))) + Float64(c0 / Float64(h * w)))) / Float64(D * D)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(c0 / Float64(w * Float64(Float64(D * D) * sqrt(Float64(h * h))))) + Float64(c0 / Float64(Float64(Float64(D * D) * h) * w)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -4e-79)
		tmp = t_0 * (((d * d) * (sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / (D * D));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-79], N[(t$95$0 * N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(c0 / N[(w * N[(N[(D * D), $MachinePrecision] * N[Sqrt[N[(h * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -4e-79

   1. Initial program 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 27.6%

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

   if -4e-79 < (*.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 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{\left(\frac{4}{2}\right)} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{2} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in D around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 16.6

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

   13. Applied rewrites 16.6%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 9: 41.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot w\right) \cdot \left(h \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \left(\left(D \cdot D\right) \cdot \sqrt{h \cdot h}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -4e-79)
     (*
      0.5
      (*
       c0
       (/
        (* (* d d) (+ (sqrt (/ (* c0 c0) (* (* h w) (* h w)))) (/ c0 (* h w))))
        (* (* D D) w))))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (+
          (/ c0 (* w (* (* D D) (sqrt (* h h)))))
          (/ c0 (* (* (* D D) h) w)))))
       (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-79) {
		tmp = 0.5 * (c0 * (((d * d) * (sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / ((D * D) * w)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -4e-79) {
		tmp = 0.5 * (c0 * (((d * d) * (Math.sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / ((D * D) * w)));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * Math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -4e-79:
		tmp = 0.5 * (c0 * (((d * d) * (math.sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / ((D * D) * w)))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -4e-79)
		tmp = Float64(0.5 * Float64(c0 * Float64(Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(Float64(h * w) * Float64(h * w)))) + Float64(c0 / Float64(h * w)))) / Float64(Float64(D * D) * w))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(c0 / Float64(w * Float64(Float64(D * D) * sqrt(Float64(h * h))))) + Float64(c0 / Float64(Float64(Float64(D * D) * h) * w)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -4e-79)
		tmp = 0.5 * (c0 * (((d * d) * (sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w)))) / ((D * D) * w)));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * ((c0 / (w * ((D * D) * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-79], N[(0.5 * N[(c0 * N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(c0 / N[(w * N[(N[(D * D), $MachinePrecision] * N[Sqrt[N[(h * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -4e-79

   1. Initial program 25.0%

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

   2. Taylor expanded in D around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 16.6%

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

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 27.0

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot w\right) \cdot \left(h \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \]

   7. Applied rewrites 27.0%

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

   if -4e-79 < (*.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 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{\left(\frac{4}{2}\right)} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{2} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in D around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 16.6

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

   13. Applied rewrites 16.6%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 10: 40.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+57}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot w\right) \cdot \left(h \cdot w\right)}} + \frac{c0}{h \cdot w}\right)\right)}{\left(D \cdot D\right) \cdot w}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{\left(D \cdot D\right) \cdot \left(w \cdot \sqrt{h \cdot h}\right)} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -1e+57)
     (*
      0.5
      (/
       (*
        c0
        (*
         (* d d)
         (+ (sqrt (/ (* c0 c0) (* (* h w) (* h w)))) (/ c0 (* h w)))))
       (* (* D D) w)))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (+
          (/ c0 (* (* D D) (* w (sqrt (* h h)))))
          (/ c0 (* (* (* D D) h) w)))))
       (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e+57) {
		tmp = 0.5 * ((c0 * ((d * d) * (sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w))))) / ((D * D) * w));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * ((c0 / ((D * D) * (w * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -1e+57) {
		tmp = 0.5 * ((c0 * ((d * d) * (Math.sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w))))) / ((D * D) * w));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * ((c0 / ((D * D) * (w * Math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	} else {
		tmp = 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -1e+57:
		tmp = 0.5 * ((c0 * ((d * d) * (math.sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w))))) / ((D * D) * w))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * ((c0 / ((D * D) * (w * math.sqrt((h * h))))) + (c0 / (((D * D) * h) * w))))
	else:
		tmp = 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -1e+57)
		tmp = Float64(0.5 * Float64(Float64(c0 * Float64(Float64(d * d) * Float64(sqrt(Float64(Float64(c0 * c0) / Float64(Float64(h * w) * Float64(h * w)))) + Float64(c0 / Float64(h * w))))) / Float64(Float64(D * D) * w)));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(c0 / Float64(Float64(D * D) * Float64(w * sqrt(Float64(h * h))))) + Float64(c0 / Float64(Float64(Float64(D * D) * h) * w)))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -1e+57)
		tmp = 0.5 * ((c0 * ((d * d) * (sqrt(((c0 * c0) / ((h * w) * (h * w)))) + (c0 / (h * w))))) / ((D * D) * w));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * ((c0 / ((D * D) * (w * sqrt((h * h))))) + (c0 / (((D * D) * h) * w))));
	else
		tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+57], N[(0.5 * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(c0 / N[(N[(D * D), $MachinePrecision] * N[(w * N[Sqrt[N[(h * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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))))) < -1.00000000000000005e57

   1. Initial program 25.0%

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

   2. Taylor expanded in D around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 16.6%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 27.0%

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

   if -1.00000000000000005e57 < (*.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 25.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 12.0

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

   7. Applied rewrites 12.0%

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

   8. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{{D}^{4} \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{\left(\frac{4}{2}\right)} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{w \cdot \sqrt{\left({D}^{2} \cdot {D}^{\left(\frac{4}{2}\right)}\right) \cdot {h}^{2}}} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in D around 0

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

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 16.6

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

   13. Applied rewrites 16.6%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 11: 40.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   2. Taylor expanded in D around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 16.6%

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

   5. Taylor expanded in d around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 27.0

          \[\leadsto 0.5 \cdot \left(c0 \cdot \frac{\left(d \cdot d\right) \cdot \left(\sqrt{\frac{c0 \cdot c0}{\left(h \cdot w\right) \cdot \left(h \cdot w\right)}} + \frac{c0}{h \cdot w}\right)}{\left(D \cdot D\right) \cdot w}\right) \]

   7. Applied rewrites 27.0%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 12: 37.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   2. Taylor expanded in D around 0

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

      1. lower-*.f64 N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 16.6%

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

   5. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 27.0%

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

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

   1. Initial program 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 13: 28.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   2. Taylor expanded in c0 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 7.8

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

   4. Applied rewrites 7.8%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 14: 27.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.0%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 8.0

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

   4. Applied rewrites 8.0%

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

   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 25.0%

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

   2. Taylor expanded in c0 around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      7. lift-*.f64 15.0

         \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   4. Applied rewrites 15.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

      5. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

      7. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

      10. lower-neg.f64 22.7

          \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

   6. Applied rewrites 22.7%

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

Alternative 15: 22.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* 0.5 (/ (* c0 (pow (* (- M) M) 0.5)) w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * pow((-M * M), 0.5)) / w);
}

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 = 0.5d0 * ((c0 * ((-m * m) ** 0.5d0)) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.pow((-M * M), 0.5)) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.pow((-M * M), 0.5)) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * (Float64(Float64(-M) * M) ^ 0.5)) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * ((-M * M) ^ 0.5)) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Power[N[((-M) * M), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   7. lift-*.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

   5. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}}}{w} \]

   7. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}}}{w} \]

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

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot {\left(\left(\mathsf{neg}\left(M\right)\right) \cdot M\right)}^{\frac{1}{2}}}{w} \]

   10. lower-neg.f64 22.7

       \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]

6. Applied rewrites 22.7%

   \[\leadsto 0.5 \cdot \frac{c0 \cdot {\left(\left(-M\right) \cdot M\right)}^{0.5}}{w} \]
7. Add Preprocessing

Alternative 16: 15.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * sqrt(-(M * M))) / w);
}

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 = 0.5d0 * ((c0 * sqrt(-(m * m))) / w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return 0.5 * ((c0 * Math.sqrt(-(M * M))) / w);
}

Python

def code(c0, w, h, D, d, M):
	return 0.5 * ((c0 * math.sqrt(-(M * M))) / w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(0.5 * Float64(Float64(c0 * sqrt(Float64(-Float64(M * M)))) / w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = 0.5 * ((c0 * sqrt(-(M * M))) / w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(0.5 * N[(N[(c0 * N[Sqrt[(-N[(M * M), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in c0 around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{\mathsf{neg}\left({M}^{2}\right)}}{w} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-{M}^{2}}}{w} \]

   6. pow2 N/A

      \[\leadsto \frac{1}{2} \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

   7. lift-*.f64 15.0

      \[\leadsto 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \]

4. Applied rewrites 15.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w}} \]
5. Add Preprocessing

Alternative 17: 0.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{2 \cdot w} \cdot \left(M \cdot \sqrt{-1}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ c0 (* 2.0 w)) (* M (sqrt -1.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (M * sqrt(-1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * (m * sqrt((-1.0d0)))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (M * Math.sqrt(-1.0));
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (M * math.sqrt(-1.0))

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(M * sqrt(-1.0)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (M * sqrt(-1.0));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(M * N[Sqrt[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in M around inf

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 0.0

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

4. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(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))))))