Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 51.6%

Time: 8.8s

Alternatives: 11

Speedup: 3.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.1% 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: 51.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 39.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 42.9

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

   8. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 42.9

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

   10. Applied rewrites 42.9%

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

   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 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   12. Applied rewrites 29.0%

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

Alternative 2: 51.1% accurate, 0.7× speedup

Math

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

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 (* (* c0 d) d)) (* (* (* h w) D) D)))
     (/ (/ (* (* (* D D) -0.25) (* (* M M) h)) d) d))))

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 * ((c0 * d) * d)) / (((h * w) * D) * D));
	} else {
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	}
	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 * ((c0 * d) * d)) / (((h * w) * D) * D));
	} else {
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	}
	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 * ((c0 * d) * d)) / (((h * w) * D) * D))
	else:
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_0 + sqrt(Float64(Float64(t_0 * t_0) - Float64(M * M))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(2.0 * Float64(Float64(c0 * d) * d)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D * D) * -0.25) * Float64(Float64(M * M) * h)) / d) / d);
	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 * ((c0 * d) * d)) / (((h * w) * D) * D));
	else
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 40.1

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

   6. Applied rewrites 40.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 40.1

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

   8. Applied rewrites 40.1%

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

   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 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   12. Applied rewrites 29.0%

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

Alternative 3: 49.7% accurate, 0.7× speedup

Math

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

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 (/ (* (* d d) (+ c0 c0)) (* (* h w) (* D D)))) (+ w w))
     (/ (/ (* (* (* D D) -0.25) (* (* M M) h)) d) d))))

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 * (((d * d) * (c0 + c0)) / ((h * w) * (D * D)))) / (w + w);
	} else {
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	}
	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 * (((d * d) * (c0 + c0)) / ((h * w) * (D * D)))) / (w + w);
	} else {
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	}
	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 * (((d * d) * (c0 + c0)) / ((h * w) * (D * D)))) / (w + w)
	else:
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d
	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(Float64(Float64(d * d) * Float64(c0 + c0)) / Float64(Float64(h * w) * Float64(D * D)))) / Float64(w + w));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D * D) * -0.25) * Float64(Float64(M * M) * h)) / d) / d);
	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 * (((d * d) * (c0 + c0)) / ((h * w) * (D * D)))) / (w + w);
	else
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	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[(N[(N[(d * d), $MachinePrecision] * N[(c0 + c0), $MachinePrecision]), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 39.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 32.6

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

   10. Applied rewrites 32.6%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   12. Applied rewrites 29.0%

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

Alternative 4: 49.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(c0, w, h, D, d, M):
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M * M))))) <= math.inf:
		tmp = (c0 / (w + w)) * ((d * d) * ((c0 + c0) / (((D * D) * h) * w)))
	else:
		tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d
	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(d * d) * Float64(Float64(c0 + c0) / Float64(Float64(Float64(D * D) * h) * w))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(D * D) * -0.25) * Float64(Float64(M * M) * h)) / d) / d);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 39.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 42.9

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

   8. Applied rewrites 42.9%

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

   10. Applied rewrites 33.1%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   12. Applied rewrites 29.0%

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

Alternative 5: 45.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 27.8

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

   10. Applied rewrites 27.8%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   12. Applied rewrites 29.0%

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

Alternative 6: 44.0% accurate, 0.7× speedup

Math

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

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))))
     (/ (/ (* (* (* D D) -0.25) (* (* M M) h)) d) d))))

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 = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	}
	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 = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	}
	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 = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d
	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(Float64(Float64(Float64(D * D) * -0.25) * Float64(Float64(M * M) * h)) / d) / d);
	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 = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
	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[(N[(N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in w around 0

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

   7. Applied rewrites 17.3%

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

   8. Taylor expanded in c0 around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 27.2

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

   10. Applied rewrites 27.2%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   12. Applied rewrites 29.0%

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

Alternative 7: 29.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\left(D \cdot D\right) \cdot -0.25\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{d} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (/ (/ (* (* (* D D) -0.25) (* (* M M) h)) d) d))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = ((((d * d) * (-0.25d0)) * ((m * m) * h)) / d_1) / d_1
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
}

Python

def code(c0, w, h, D, d, M):
	return ((((D * D) * -0.25) * ((M * M) * h)) / d) / d

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(Float64(Float64(D * D) * -0.25) * Float64(Float64(M * M) * h)) / d) / d)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((((D * D) * -0.25) * ((M * M) * h)) / d) / d;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in w around 0

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

7. Applied rewrites 17.3%

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

8. Taylor expanded in c0 around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 27.2

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

10. Applied rewrites 27.2%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

12. Applied rewrites 29.0%

    \[\leadsto \frac{\frac{\left(\left(D \cdot D\right) \cdot -0.25\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}}{d} \]
13. Add Preprocessing

Alternative 8: 28.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(D \cdot D\right) \cdot -0.25}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{d} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ (* (* D D) -0.25) d) (/ (* (* M M) h) d)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (((D * D) * -0.25) / d) * (((M * M) * h) / d);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (((d * d) * (-0.25d0)) / d_1) * (((m * m) * h) / d_1)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (((D * D) * -0.25) / d) * (((M * M) * h) / d);
}

Python

def code(c0, w, h, D, d, M):
	return (((D * D) * -0.25) / d) * (((M * M) * h) / d)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(Float64(D * D) * -0.25) / d) * Float64(Float64(Float64(M * M) * h) / d))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (((D * D) * -0.25) / d) * (((M * M) * h) / d);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in w around 0

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

7. Applied rewrites 17.3%

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

8. Taylor expanded in c0 around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 27.2

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

10. Applied rewrites 27.2%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. times-frac N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 N/A

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

    11. pow2 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. pow2 N/A

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

    15. lift-*.f64 N/A

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

    16. pow2 N/A

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

    17. lower-/.f64 N/A

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

    18. pow2 N/A

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

    19. lift-*.f64 N/A

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

    20. lift-*.f64 28.9

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

12. Applied rewrites 28.9%

    \[\leadsto \frac{\left(D \cdot D\right) \cdot -0.25}{d} \cdot \frac{\left(M \cdot M\right) \cdot h}{\color{blue}{d}} \]
13. Add Preprocessing

Alternative 9: 28.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d \cdot d} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (/ (* (* -0.25 (* D D)) (* M (* M h))) (* d d)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((-0.25 * (D * D)) * (M * (M * h))) / (d * d);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (((-0.25d0) * (d * d)) * (m * (m * h))) / (d_1 * d_1)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((-0.25 * (D * D)) * (M * (M * h))) / (d * d);
}

Python

def code(c0, w, h, D, d, M):
	return ((-0.25 * (D * D)) * (M * (M * h))) / (d * d)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(-0.25 * Float64(D * D)) * Float64(M * Float64(M * h))) / Float64(d * d))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((-0.25 * (D * D)) * (M * (M * h))) / (d * d);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(-0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(M * N[(M * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in w around 0

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

7. Applied rewrites 17.3%

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

8. Taylor expanded in c0 around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 27.2

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

10. Applied rewrites 27.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 28.4

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

12. Applied rewrites 28.4%

    \[\leadsto \frac{\left(-0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)}{d \cdot d} \]
13. Add Preprocessing

Alternative 10: 27.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(D \cdot D\right) \cdot -0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{h}{d \cdot d}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (* (* D D) -0.25) (* (* M M) (/ h (* d d)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((D * D) * -0.25) * ((M * M) * (h / (d * d)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = ((d * d) * (-0.25d0)) * ((m * m) * (h / (d_1 * d_1)))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((D * D) * -0.25) * ((M * M) * (h / (d * d)));
}

Python

def code(c0, w, h, D, d, M):
	return ((D * D) * -0.25) * ((M * M) * (h / (d * d)))

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(D * D) * -0.25) * Float64(Float64(M * M) * Float64(h / Float64(d * d))))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((D * D) * -0.25) * ((M * M) * (h / (d * d)));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(D * D), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * N[(h / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in w around 0

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

7. Applied rewrites 17.3%

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

8. Taylor expanded in c0 around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 27.2

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

10. Applied rewrites 27.2%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. pow2 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. associate-/l* N/A

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

    10. pow2 N/A

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

    11. lower-*.f64 N/A

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

    12. pow2 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f64 N/A

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

    15. pow2 N/A

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

    16. lift-*.f64 N/A

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

    17. associate-/l* N/A

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

    18. lower-*.f64 N/A

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

    19. pow2 N/A

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

    20. lift-*.f64 N/A

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

    21. lower-/.f64 N/A

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

    22. pow2 N/A

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

    23. lift-*.f64 27.0

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

12. Applied rewrites 27.0%

    \[\leadsto \left(\left(D \cdot D\right) \cdot -0.25\right) \cdot \left(\left(M \cdot M\right) \cdot \color{blue}{\frac{h}{d \cdot d}}\right) \]
13. Add Preprocessing

Alternative 11: 0.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{c0 \cdot \left(\sqrt{-1} \cdot M\right)}{w + w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (/ (* c0 (* (sqrt -1.0) M)) (+ w w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 * (sqrt(-1.0) * M)) / (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 * (sqrt((-1.0d0)) * m)) / (w + w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 * (Math.sqrt(-1.0) * M)) / (w + w);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 * (math.sqrt(-1.0) * M)) / (w + w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 * Float64(sqrt(-1.0) * M)) / Float64(w + w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 * (sqrt(-1.0) * M)) / (w + w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 * N[(N[Sqrt[-1.0], $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in c0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 0.0

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

4. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

6. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\frac{c0 \cdot \left(\sqrt{-1} \cdot M\right)}{w + w}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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))))))