Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.6% → 55.2%

Time: 8.2s

Alternatives: 9

Speedup: 5.5×

• could not determine a ground truth

  1. c0 = 3.131514779801691e+300
  2. w = 1.0335070387674888e-300
  3. h = -8.672239276703662e-301
  4. D = 7.015802693061219e-211
  5. d = -1.6510581491109855e+270
  6. M = -9.690717914848775e+23

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: 25.6% 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: 55.2% accurate, 0.7× speedup

Math

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

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $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 76.2%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.2

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

   6. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 80.3

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

   10. Applied rewrites 80.3%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 42.5%

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

Alternative 2: 54.8% accurate, 0.7× speedup

Math

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

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(d * N[(d * c0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $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 76.2%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.2

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

   6. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 79.2

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

   8. Applied rewrites 79.2%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 42.5%

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

Alternative 3: 53.1% accurate, 0.7× speedup

Math

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

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 (* (* (/ (* d d) (* (* (* D w) D) h)) c0) 2.0)) (+ w w))
     (/ (* (* 0.25 (* D D)) (* (* M M) h)) (* d d)))))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * N[(N[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * w), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $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 76.2%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 79.2

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

   6. Applied rewrites 79.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

   10. Applied rewrites 74.0%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 42.5%

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

Alternative 4: 50.4% accurate, 0.7× speedup

Math

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

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] * c0), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(N[(w * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $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 76.2%

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   6. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 63.0%

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

   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%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 42.5%

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

Alternative 5: 49.9% accurate, 0.7× speedup

Math

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

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 * c0), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(N[(D * w), $MachinePrecision] * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $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 76.2%

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   6. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. associate-*l* N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. pow2 N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 64.6

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

   8. Applied rewrites 64.6%

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

   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%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 42.5%

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

Alternative 6: 49.4% accurate, 0.7× speedup

Math

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

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(c0 * N[(N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $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 76.2%

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in c0 around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.7

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

   6. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

   8. Applied rewrites 66.0%

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

   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%

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

   3. Applied rewrites 4.5%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 1.5%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 42.5%

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

Alternative 7: 33.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;D \leq 1.1 \cdot 10^{+149}:\\ \;\;\;\;\frac{\left(0.25 \cdot \left(D \cdot D\right)\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (if (<= D 1.1e+149)
   (/ (* (* 0.25 (* D D)) (* (* M M) h)) (* d d))
   (* (/ (* (* c0 c0) 0.0) w) -0.5)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double tmp;
	if (D <= 1.1e+149) {
		tmp = ((0.25 * (D * D)) * ((M * M) * h)) / (d * d);
	} else {
		tmp = (((c0 * c0) * 0.0) / w) * -0.5;
	}
	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 (d <= 1.1d+149) then
        tmp = ((0.25d0 * (d * d)) * ((m * m) * h)) / (d_1 * d_1)
    else
        tmp = (((c0 * c0) * 0.0d0) / w) * (-0.5d0)
    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 (D <= 1.1e+149) {
		tmp = ((0.25 * (D * D)) * ((M * M) * h)) / (d * d);
	} else {
		tmp = (((c0 * c0) * 0.0) / w) * -0.5;
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	tmp = 0
	if D <= 1.1e+149:
		tmp = ((0.25 * (D * D)) * ((M * M) * h)) / (d * d)
	else:
		tmp = (((c0 * c0) * 0.0) / w) * -0.5
	return tmp

Julia

function code(c0, w, h, D, d, M)
	tmp = 0.0
	if (D <= 1.1e+149)
		tmp = Float64(Float64(Float64(0.25 * Float64(D * D)) * Float64(Float64(M * M) * h)) / Float64(d * d));
	else
		tmp = Float64(Float64(Float64(Float64(c0 * c0) * 0.0) / w) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	tmp = 0.0;
	if (D <= 1.1e+149)
		tmp = ((0.25 * (D * D)) * ((M * M) * h)) / (d * d);
	else
		tmp = (((c0 * c0) * 0.0) / w) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := If[LessEqual[D, 1.1e+149], N[(N[(N[(0.25 * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c0 * c0), $MachinePrecision] * 0.0), $MachinePrecision] / w), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if D < 1.1e149

   1. Initial program 26.9%

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

   3. Applied rewrites 28.1%

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

   4. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   6. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\left(c0 \cdot c0\right) \cdot \mathsf{fma}\left(-0.5, \frac{0 \cdot \frac{d \cdot d}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}}{w}, \frac{0.25 \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot h\right)\right)}{c0 \cdot \left(c0 \cdot \left(d \cdot d\right)\right)}\right)} \]

   7. Taylor expanded in c0 around 0

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}} \]
   8. 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

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

   9. Applied rewrites 34.8%

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

   if 1.1e149 < D

   1. Initial program 2.3%

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

   2. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 26.0

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

   4. Applied rewrites 26.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 28.0

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

   6. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 38.5

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

   8. Applied rewrites 38.5%

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

   9. Taylor expanded in c0 around -inf

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

   11. Applied rewrites 15.8%

       \[\leadsto \color{blue}{\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{2 \cdot w} \cdot \left(-0 \cdot c0\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* (/ c0 (* 2.0 w)) (- (* 0.0 c0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * -(0.0 * c0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * -(0.0d0 * c0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * -(0.0 * c0);
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * -(0.0 * c0)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(-Float64(0.0 * c0)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * -(0.0 * c0);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * (-N[(0.0 * c0), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 25.6%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.4

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

4. Applied rewrites 36.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 40.0

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

6. Applied rewrites 40.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 42.6

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

8. Applied rewrites 42.6%

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

9. Taylor expanded in c0 around -inf

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

    1. pow2 N/A

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

    2. pow2 N/A

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

    3. frac-times N/A

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

    4. pow2 N/A

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

    5. pow2 N/A

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

11. Applied rewrites 29.4%

    \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(-0 \cdot c0\right)} \]
12. Add Preprocessing

Alternative 9: 25.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5 \end{array} \]

FPCore

(FPCore (c0 w h D d M) :precision binary64 (* (/ (* (* c0 c0) 0.0) w) -0.5))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (((c0 * c0) * 0.0) / w) * -0.5;
}

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 * c0) * 0.0d0) / w) * (-0.5d0)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (((c0 * c0) * 0.0) / w) * -0.5;
}

Python

def code(c0, w, h, D, d, M):
	return (((c0 * c0) * 0.0) / w) * -0.5

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(Float64(c0 * c0) * 0.0) / w) * -0.5)
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (((c0 * c0) * 0.0) / w) * -0.5;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[(c0 * c0), $MachinePrecision] * 0.0), $MachinePrecision] / w), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 25.6%

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

2. Taylor expanded in c0 around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 36.4

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

4. Applied rewrites 36.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 40.0

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

6. Applied rewrites 40.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 42.6

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

8. Applied rewrites 42.6%

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

9. Taylor expanded in c0 around -inf

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

11. Applied rewrites 25.0%

    \[\leadsto \color{blue}{\frac{\left(c0 \cdot c0\right) \cdot 0}{w} \cdot -0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025113 
(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))))))