Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.3% → 54.6%

Time: 9.6s

Alternatives: 6

Speedup: 7.8×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 54.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{{\left(d \cdot c0\right)}^{2} \cdot 2}{D \cdot \left(w \cdot h\right)}}{D}}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \end{array} \]

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)
     (/ (/ (/ (* (pow (* d c0) 2.0) 2.0) (* D (* w h))) D) (* w 2.0))
     (/ (* c0 0.0) (+ w 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 = (((pow((d * c0), 2.0) * 2.0) / (D * (w * h))) / D) / (w * 2.0);
	} else {
		tmp = (c0 * 0.0) / (w + 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 = (((Math.pow((d * c0), 2.0) * 2.0) / (D * (w * h))) / D) / (w * 2.0);
	} else {
		tmp = (c0 * 0.0) / (w + 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 = (((math.pow((d * c0), 2.0) * 2.0) / (D * (w * h))) / D) / (w * 2.0)
	else:
		tmp = (c0 * 0.0) / (w + 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(Float64(Float64(Float64((Float64(d * c0) ^ 2.0) * 2.0) / Float64(D * Float64(w * h))) / D) / Float64(w * 2.0));
	else
		tmp = Float64(Float64(c0 * 0.0) / Float64(w + 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 = (((((d * c0) ^ 2.0) * 2.0) / (D * (w * h))) / D) / (w * 2.0);
	else
		tmp = (c0 * 0.0) / (w + 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[(N[(N[(N[(N[Power[N[(d * c0), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(w + 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 70.7%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 77.9

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

   10. Applied rewrites 77.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 80.3

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-pow.f64 N/A

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

       9. unpow-prod-down N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. *-commutative N/A

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

       14. unpow-prod-down N/A

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

       15. lift-pow.f64 N/A

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

       16. lift-*.f64 80.3

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

       20. lower-*.f64 N/A

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

       21. *-commutative N/A

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

       22. lower-*.f64 80.3

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

   12. Applied rewrites 80.3%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w \cdot 2}} \]

       2. *-commutative N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{2 \cdot w}} \]

       3. count-2-rev N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

       4. lower-+.f64 45.0

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

   12. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 54.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot \left(d \cdot \frac{c0}{\left(D \cdot \left(w \cdot h\right)\right) \cdot D}\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \end{array} \]

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 (* (* d (* d (/ c0 (* (* D (* w h)) D)))) 2.0))
     (/ (* c0 0.0) (+ w 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 * ((d * (d * (c0 / ((D * (w * h)) * D)))) * 2.0);
	} else {
		tmp = (c0 * 0.0) / (w + 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 * ((d * (d * (c0 / ((D * (w * h)) * D)))) * 2.0);
	} else {
		tmp = (c0 * 0.0) / (w + 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 * ((d * (d * (c0 / ((D * (w * h)) * D)))) * 2.0)
	else:
		tmp = (c0 * 0.0) / (w + 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(Float64(d * Float64(d * Float64(c0 / Float64(Float64(D * Float64(w * h)) * D)))) * 2.0));
	else
		tmp = Float64(Float64(c0 * 0.0) / Float64(w + 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 * ((d * (d * (c0 / ((D * (w * h)) * D)))) * 2.0);
	else
		tmp = (c0 * 0.0) / (w + 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[(d * N[(d * N[(c0 / N[(N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(w + 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 70.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

   9. Applied rewrites 75.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. pow2 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. *-commutative N/A

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

       15. pow2 N/A

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

       16. associate-*l* N/A

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

       17. lift-*.f64 N/A

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

       18. lift-*.f64 N/A

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

       19. lift-*.f64 79.1

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

   11. Applied rewrites 79.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w \cdot 2}} \]

       2. *-commutative N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{2 \cdot w}} \]

       3. count-2-rev N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

       4. lower-+.f64 45.0

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

   12. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot \left(d \cdot \frac{c0}{\left(D \cdot \left(w \cdot h\right)\right) \cdot D}\right)\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(d \cdot \left(d \cdot c0\right)\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \end{array} \]

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)
     (* (/ c0 (+ w w)) (/ (* 2.0 (* d (* d c0))) (* (* (* h w) D) D)))
     (/ (* c0 0.0) (+ w 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 = (c0 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * 0.0) / (w + 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 = (c0 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * 0.0) / (w + 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 = (c0 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D))
	else:
		tmp = (c0 * 0.0) / (w + 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(Float64(c0 / Float64(w + w)) * Float64(Float64(2.0 * Float64(d * Float64(d * c0))) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(c0 * 0.0) / Float64(w + 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 = (c0 / (w + w)) * ((2.0 * (d * (d * c0))) / (((h * w) * D) * D));
	else
		tmp = (c0 * 0.0) / (w + 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[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(w + 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 70.7%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 75.5

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

   9. Applied rewrites 75.5%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w \cdot 2}} \]

       2. *-commutative N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{2 \cdot w}} \]

       3. count-2-rev N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

       4. lower-+.f64 45.0

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

   12. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(d \cdot \left(d \cdot c0\right)\right)}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \end{array} \]

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)
     (/ (* (* d c0) (* d c0)) (* (* (* D D) h) (* w w)))
     (/ (* c0 0.0) (+ w 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 = ((d * c0) * (d * c0)) / (((D * D) * h) * (w * w));
	} else {
		tmp = (c0 * 0.0) / (w + 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 = ((d * c0) * (d * c0)) / (((D * D) * h) * (w * w));
	} else {
		tmp = (c0 * 0.0) / (w + 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 = ((d * c0) * (d * c0)) / (((D * D) * h) * (w * w))
	else:
		tmp = (c0 * 0.0) / (w + 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(Float64(Float64(d * c0) * Float64(d * c0)) / Float64(Float64(Float64(D * D) * h) * Float64(w * w)));
	else
		tmp = Float64(Float64(c0 * 0.0) / Float64(w + 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 = ((d * c0) * (d * c0)) / (((D * D) * h) * (w * w));
	else
		tmp = (c0 * 0.0) / (w + 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[(N[(N[(d * c0), $MachinePrecision] * N[(d * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(w + 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 70.7%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 66.0

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

   9. Applied rewrites 66.0%

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

   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 0.0%

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w \cdot 2}} \]

       2. *-commutative N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{2 \cdot w}} \]

       3. count-2-rev N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

       4. lower-+.f64 45.0

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

   12. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 47.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_0 + \sqrt{t\_0 \cdot t\_0 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot 0}{w + w}\\ \end{array} \end{array} \]

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)
     (* (* c0 c0) (/ (* d d) (* (* (* D D) h) (* w w))))
     (/ (* c0 0.0) (+ w 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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)));
	} else {
		tmp = (c0 * 0.0) / (w + 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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)));
	} else {
		tmp = (c0 * 0.0) / (w + 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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)))
	else:
		tmp = (c0 * 0.0) / (w + 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(Float64(c0 * c0) * Float64(Float64(d * d) / Float64(Float64(Float64(D * D) * h) * Float64(w * w))));
	else
		tmp = Float64(Float64(c0 * 0.0) / Float64(w + 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 = (c0 * c0) * ((d * d) / (((D * D) * h) * (w * w)));
	else
		tmp = (c0 * 0.0) / (w + 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[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(w + 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 70.7%

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. unpow-prod-down N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-prod-down N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

   9. Applied rewrites 57.0%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 2.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 2.6%

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

   8. Taylor expanded in c0 around 0

      \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w \cdot 2}} \]

       2. *-commutative N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{2 \cdot w}} \]

       3. count-2-rev N/A

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

       4. lower-+.f64 45.0

          \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

   12. Applied rewrites 45.0%

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 33.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \frac{c0 \cdot 0}{w + w} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 * 0.0) / (w + 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 = (c0 * 0.0d0) / (w + w)
end function

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 * 0.0) / Float64(w + w))
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 * 0.0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.5%

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

4. Applied rewrites 23.6%

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

5. Taylor expanded in c0 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

7. Applied rewrites 4.1%

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

8. Taylor expanded in c0 around 0

   \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
9. Step-by-step derivation

10. Applied rewrites 34.7%

    \[\leadsto \frac{c0 \cdot 0}{w \cdot 2} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w \cdot 2}} \]

    2. *-commutative N/A

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{2 \cdot w}} \]

    3. count-2-rev N/A

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

    4. lower-+.f64 34.7

       \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]

12. Applied rewrites 34.7%

    \[\leadsto \frac{c0 \cdot 0}{\color{blue}{w + w}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))