Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.0% → 55.9%

Time: 10.5s

Alternatives: 11

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 80.9

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

   9. Applied rewrites 80.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. count-2-rev 41.5

         \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot \color{blue}{h}\right) \cdot -0.25}{d \cdot d} \]

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

   10. Applied rewrites 43.1%

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

Alternative 2: 55.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 80.9

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

   9. Applied rewrites 80.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

   12. Applied rewrites 43.1%

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

Alternative 3: 54.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 80.9

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

   9. Applied rewrites 80.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{\left(2 \cdot \left(d \cdot c0\right)\right) \cdot d}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{\left(h \cdot \left(w \cdot D\right)\right) \cdot D}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 79.5

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

   11. Applied rewrites 79.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \frac{2 \cdot \left(\left(d \cdot c0\right) \cdot d\right)}{h \cdot \left(\left(D \cdot w\right) \cdot D\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.5% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.7

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

   7. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lower-+.f64 78.3

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

   11. Applied rewrites 78.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{c0}{w + w} \cdot \frac{2 \cdot \left(\left(d \cdot d\right) \cdot c0\right)}{h \cdot \left(\left(D \cdot w\right) \cdot D\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.1% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

   7. Applied rewrites 69.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 69.5

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

   9. Applied rewrites 69.5%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 69.5

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

   11. Applied rewrites 69.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{\left(D \cdot w\right) \cdot \left(D \cdot w\right)}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.7% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.5

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

   7. Applied rewrites 62.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\left(d \cdot c0\right) \cdot \left(d \cdot c0\right)}{D \cdot D}}{\left(w \cdot w\right) \cdot h}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.7% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.9%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. associate-/l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   7. Applied rewrites 56.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

   6. Taylor expanded in c0 around 0

      \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   7. 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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow-prod-down N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 41.5

          \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

   8. Applied rewrites 41.5%

      \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 41.5

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

   10. Applied rewrites 41.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-*.f64 41.6

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

   12. Applied rewrites 41.6%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty:\\ \;\;\;\;\left(c0 \cdot c0\right) \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot \left(w \cdot w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot h\right)\right) \cdot -0.25}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (((M * D) * ((M * D) * h)) * -0.25) / (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 = (((m * d) * ((m * d) * h)) * (-0.25d0)) / (d_1 * d_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in w around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 20.1%

   \[\leadsto \color{blue}{\frac{\frac{{\left(c0 \cdot d\right)}^{2}}{\left(D \cdot D\right) \cdot h} + \frac{{\left(D \cdot M\right)}^{2} \cdot \left(\left(w \cdot w\right) \cdot h\right)}{d \cdot d} \cdot -0.25}{w \cdot w}} \]

6. Taylor expanded in c0 around 0

   \[\leadsto \frac{-1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
7. 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. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. unpow-prod-down N/A

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

   7. lower-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 31.8

       \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{d \cdot d} \]

8. Applied rewrites 31.8%

   \[\leadsto \frac{\left({\left(M \cdot D\right)}^{2} \cdot h\right) \cdot -0.25}{\color{blue}{d \cdot d}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 31.8

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

10. Applied rewrites 31.8%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*l* N/A

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

    6. lower-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lift-*.f64 31.9

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

12. Applied rewrites 31.9%

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

13. Final simplification 31.9%

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

Alternative 11: 0.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (w + w)) * (sqrt((-1.0d0)) * m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in c0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025058 
(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))))))