Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.4% → 68.7%

Time: 14.5s

Alternatives: 8

Speedup: 156.0×

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: 25.4% 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: 68.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_0 + Math.sqrt(((t_0 * t_0) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 * (((2.0 * ((d * d) * c0)) / ((D * D) * w)) / (w * h))) * 0.5;
	} else {
		tmp = (h * ((D / d) * (M * ((D / d) * M)))) * 0.25;
	}
	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 * (((2.0 * ((d * d) * c0)) / ((D * D) * w)) / (w * h))) * 0.5
	else:
		tmp = (h * ((D / d) * (M * ((D / d) * M)))) * 0.25
	return tmp

Julia

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

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_0 + sqrt(((t_0 * t_0) - (M * M))))) <= Inf)
		tmp = (c0 * (((2.0 * ((d * d) * c0)) / ((D * D) * w)) / (w * h))) * 0.5;
	else
		tmp = (h * ((D / d) * (M * ((D / d) * M)))) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in h around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 78.3%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 73.7%

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

Alternative 2: 67.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

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

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in c0 around inf

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

   8. Applied rewrites 67.5%

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

   10. Applied rewrites 74.7%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 73.7%

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

4. Final simplification 74.0%

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

Alternative 3: 66.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

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

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in c0 around inf

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

   8. Applied rewrites 67.5%

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

   10. Applied rewrites 67.8%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 73.7%

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

4. Final simplification 71.9%

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

Alternative 4: 65.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

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

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in c0 around inf

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

   8. Applied rewrites 67.5%

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

   10. Applied rewrites 67.6%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 73.7%

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

4. Final simplification 71.8%

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

Alternative 5: 63.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 71.5%

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

   11. Applied rewrites 73.7%

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

4. Final simplification 71.4%

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

Alternative 6: 54.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 22.4%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

   9. Applied rewrites 71.5%

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

   10. Taylor expanded in D around 0

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

   12. Applied rewrites 55.0%

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

4. Final simplification 58.4%

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

Alternative 7: 42.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \cdot M \leq 5 \cdot 10^{-237}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(h \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= (* M M) 5e-237) 0.0 (* (* h (/ (* (* D M) (* D M)) (* d d))) 0.25)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 5e-237) {
		tmp = 0.0;
	} else {
		tmp = (h * (((D * M) * (D * M)) / (d * d))) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m * m) <= 5d-237) then
        tmp = 0.0d0
    else
        tmp = (h * (((d * m) * (d * m)) / (d_1 * d_1))) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if ((M * M) <= 5e-237) {
		tmp = 0.0;
	} else {
		tmp = (h * (((D * M) * (D * M)) / (d * d))) * 0.25;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if (M * M) <= 5e-237:
		tmp = 0.0
	else:
		tmp = (h * (((D * M) * (D * M)) / (d * d))) * 0.25
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (Float64(M * M) <= 5e-237)
		tmp = 0.0;
	else
		tmp = Float64(Float64(h * Float64(Float64(Float64(D * M) * Float64(D * M)) / Float64(d * d))) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if ((M * M) <= 5e-237)
		tmp = 0.0;
	else
		tmp = (h * (((D * M) * (D * M)) / (d * d))) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[N[(M * M), $MachinePrecision], 5e-237], 0.0, N[(N[(h * N[(N[(N[(D * M), $MachinePrecision] * N[(D * M), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M M) < 5.0000000000000002e-237

   1. Initial program 24.6%

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

   3. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

   5. Applied rewrites 40.6%

      \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 57.9%

      \[\leadsto \color{blue}{0} \]

   if 5.0000000000000002e-237 < (*.f64 M M)

   1. Initial program 21.7%

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

   4. Applied rewrites 37.8%

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

   5. Taylor expanded in c0 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   9. Applied rewrites 52.5%

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

   10. Taylor expanded in D around 0

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

   12. Applied rewrites 42.1%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \leq 5 \cdot 10^{-237}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(h \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d \cdot d}\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 32.8% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.0d0
end function

Java

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

Python

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

Julia

function code(c0, w, h, D, d, M)
	return 0.0
end

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := 0.0

Derivation

1. Initial program 22.9%

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

3. Taylor expanded in c0 around -inf

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

   1. *-commutative N/A

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

5. Applied rewrites 28.2%

   \[\leadsto \color{blue}{\frac{0 \cdot \left(c0 \cdot c0\right)}{w} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 37.8%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 37.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024364 
(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))))))