Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 46.5%

Time: 8.7s

Alternatives: 5

Speedup: 7.8×

• could not determine a ground truth

  1. c0 = -8.678827219488743e+277
  2. w = -1.737277860584845e-247
  3. h = -2.5143218391237853e-291
  4. D = -1.4676877907945415e-300
  5. d = -5.9428016356918115e+280
  6. M = -9.962335369065215e+142

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% 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: 46.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 9.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{c0 \cdot 0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(d \cdot \left(d \cdot \frac{\frac{c0}{D \cdot \left(w \cdot h\right)}}{D}\right)\right) \cdot 2\right)}{w \cdot 2}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (if (<= M_m 9.5e-180)
   (/ (* c0 0.0) (* 2.0 w))
   (/ (* c0 (* (* d (* d (/ (/ c0 (* D (* w h))) D))) 2.0)) (* w 2.0))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.5e-180) {
		tmp = (c0 * 0.0) / (2.0 * w);
	} else {
		tmp = (c0 * ((d * (d * ((c0 / (D * (w * h))) / D))) * 2.0)) / (w * 2.0);
	}
	return tmp;
}

Fortran

M_m =     private
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_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_m
    real(8) :: tmp
    if (m_m <= 9.5d-180) then
        tmp = (c0 * 0.0d0) / (2.0d0 * w)
    else
        tmp = (c0 * ((d_1 * (d_1 * ((c0 / (d * (w * h))) / d))) * 2.0d0)) / (w * 2.0d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double tmp;
	if (M_m <= 9.5e-180) {
		tmp = (c0 * 0.0) / (2.0 * w);
	} else {
		tmp = (c0 * ((d * (d * ((c0 / (D * (w * h))) / D))) * 2.0)) / (w * 2.0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	tmp = 0
	if M_m <= 9.5e-180:
		tmp = (c0 * 0.0) / (2.0 * w)
	else:
		tmp = (c0 * ((d * (d * ((c0 / (D * (w * h))) / D))) * 2.0)) / (w * 2.0)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_m)
	tmp = 0.0
	if (M_m <= 9.5e-180)
		tmp = Float64(Float64(c0 * 0.0) / Float64(2.0 * w));
	else
		tmp = Float64(Float64(c0 * Float64(Float64(d * Float64(d * Float64(Float64(c0 / Float64(D * Float64(w * h))) / D))) * 2.0)) / Float64(w * 2.0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_m)
	tmp = 0.0;
	if (M_m <= 9.5e-180)
		tmp = (c0 * 0.0) / (2.0 * w);
	else
		tmp = (c0 * ((d * (d * ((c0 / (D * (w * h))) / D))) * 2.0)) / (w * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := If[LessEqual[M$95$m, 9.5e-180], N[(N[(c0 * 0.0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[(N[(d * N[(d * N[(N[(c0 / N[(D * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(w * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 9.49999999999999934e-180

   1. Initial program 29.1%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 31.4%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 35.0%

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

   if 9.49999999999999934e-180 < M

   1. Initial program 30.4%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 46.4

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

   5. Applied rewrites 46.4%

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

   7. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   9. Applied rewrites 49.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 56.4

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/r* N/A

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

       11. lower-/.f64 N/A

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

       12. lower-/.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. lift-*.f64 60.3

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

   11. Applied rewrites 60.3%

       \[\leadsto \frac{c0 \cdot \left(\left(d \cdot \left(d \cdot \frac{\frac{c0}{D \cdot \left(w \cdot h\right)}}{D}\right)\right) \cdot 2\right)}{w \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 9.5 \cdot 10^{-180}:\\ \;\;\;\;\frac{c0 \cdot 0}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left(d \cdot \left(d \cdot \frac{\frac{c0}{D \cdot \left(w \cdot h\right)}}{D}\right)\right) \cdot 2\right)}{w \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.1% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w))) (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (* t_0 (/ (* 2.0 (* (* c0 d) d)) (* (* (* h w) D) D)))
     (/ (* c0 0.0) (* 2.0 w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * ((2.0 * ((c0 * d) * d)) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * 0.0) / (2.0 * w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((2.0 * ((c0 * d) * d)) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * 0.0) / (2.0 * w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = t_0 * ((2.0 * ((c0 * d) * d)) / (((h * w) * D) * D))
	else:
		tmp = (c0 * 0.0) / (2.0 * w)
	return tmp

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(2.0 * N[(N[(c0 * d), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 77.0

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

   7. Applied rewrites 77.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 40.7%

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

4. Final simplification 54.4%

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

Alternative 3: 55.3% accurate, 0.7× speedup

Math

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

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_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 M_m)))))
        INFINITY)
     (* (/ c0 (+ w w)) (/ (* 2.0 (* (* d d) c0)) (* (* (* h w) D) D)))
     (/ (* c0 0.0) (* 2.0 w)))))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_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 * M_m))))) <= ((double) INFINITY)) {
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * 0.0) / (2.0 * w);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_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 * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D));
	} else {
		tmp = (c0 * 0.0) / (2.0 * w);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_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 * M_m))))) <= math.inf:
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D))
	else:
		tmp = (c0 * 0.0) / (2.0 * w)
	return tmp

Julia

M_m = abs(M)
function code(c0, w, h, D, d, M_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 * M_m))))) <= Inf)
		tmp = Float64(Float64(c0 / Float64(w + w)) * Float64(Float64(2.0 * Float64(Float64(d * d) * c0)) / Float64(Float64(Float64(h * w) * D) * D)));
	else
		tmp = Float64(Float64(c0 * 0.0) / Float64(2.0 * w));
	end
	return tmp
end

MATLAB

M_m = abs(M);
function tmp_2 = code(c0, w, h, D, d, M_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 * M_m))))) <= Inf)
		tmp = (c0 / (w + w)) * ((2.0 * ((d * d) * c0)) / (((h * w) * D) * D));
	else
		tmp = (c0 * 0.0) / (2.0 * w);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$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$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(N[(d * d), $MachinePrecision] * c0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(h * w), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * 0.0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.2%

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

   3. Taylor expanded in c0 around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 76.0

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

   7. Applied rewrites 76.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c0 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in c0 around 0

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

   8. Applied rewrites 35.8%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 40.7%

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

4. Final simplification 54.1%

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

Alternative 4: 33.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \frac{c0 \cdot 0}{2 \cdot w} \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 (/ (* c0 0.0) (* 2.0 w)))

C

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

Fortran

M_m =     private
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_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_m
    code = (c0 * 0.0d0) / (2.0d0 * w)
end function

Java

M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D, double d, double M_m) {
	return (c0 * 0.0) / (2.0 * w);
}

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return (c0 * 0.0) / (2.0 * w)

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(N[(c0 * 0.0), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in c0 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 4.7%

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

6. Taylor expanded in c0 around 0

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

8. Applied rewrites 26.0%

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

   1. lift-*.f64 N/A

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

10. Applied rewrites 29.1%

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

11. Final simplification 29.1%

    \[\leadsto \frac{c0 \cdot 0}{2 \cdot w} \]
12. Add Preprocessing

Alternative 5: 29.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ \frac{c0}{w + w} \cdot 0 \end{array} \]

FPCore

M_m = (fabs.f64 M)
(FPCore (c0 w h D d M_m) :precision binary64 (* (/ c0 (+ w w)) 0.0))

C

M_m = fabs(M);
double code(double c0, double w, double h, double D, double d, double M_m) {
	return (c0 / (w + w)) * 0.0;
}

Fortran

M_m =     private
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_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_m
    code = (c0 / (w + w)) * 0.0d0
end function

Java

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

Python

M_m = math.fabs(M)
def code(c0, w, h, D, d, M_m):
	return (c0 / (w + w)) * 0.0

Julia

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

MATLAB

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

Wolfram

M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D_, d_, M$95$m_] := N[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in c0 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

5. Applied rewrites 4.7%

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

6. Taylor expanded in c0 around 0

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

8. Applied rewrites 26.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 26.0

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

10. Applied rewrites 26.0%

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

11. Final simplification 26.0%

    \[\leadsto \frac{c0}{w + w} \cdot 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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))))))