Henrywood and Agarwal, Equation (13)

Percentage Accurate: 25.1% → 44.7%

Time: 10.1s

Alternatives: 12

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 44.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{d\_m}{\left(h \cdot w\right) \cdot D\_m} \cdot \frac{d\_m}{D\_m}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, t\_0, M\_m\right)\right)}^{0.5}, \frac{d\_m \cdot \sqrt{\frac{c0}{h \cdot w}}}{D\_m}, c0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M\_m \cdot M\_m\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (* (/ d_m (* (* h w) D_m)) (/ d_m D_m)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m))))) INFINITY)
     (*
      t_1
      (fma
       (pow (fma c0 t_0 M_m) 0.5)
       (/ (* d_m (sqrt (/ c0 (* h w)))) D_m)
       (* c0 t_0)))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = (d_m / ((h * w) * D_m)) * (d_m / D_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(pow(fma(c0, t_0, M_m), 0.5), ((d_m * sqrt((c0 / (h * w)))) / D_m), (c0 * t_0));
	} else {
		tmp = ((pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(Float64(d_m / Float64(Float64(h * w) * D_m)) * Float64(d_m / D_m))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_1 * fma((fma(c0, t_0, M_m) ^ 0.5), Float64(Float64(d_m * sqrt(Float64(c0 / Float64(h * w)))) / D_m), Float64(c0 * t_0)));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(N[(d$95$m / N[(N[(h * w), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[Power[N[(c0 * t$95$0 + M$95$m), $MachinePrecision], 0.5], $MachinePrecision] * N[(N[(d$95$m * N[Sqrt[N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / D$95$m), $MachinePrecision] + N[(c0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 32.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]

   5. Applied rewrites 32.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 33.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}\right) - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 38.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{0.5}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}\right)\right) \]

   9. Applied rewrites 38.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{0.5}, c0 \cdot \color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)}\right) \]

   10. Taylor expanded in D around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 34.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, \frac{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}}}{D}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

   12. Applied rewrites 34.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, \color{blue}{\frac{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}}}{D}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

   13. Taylor expanded in d around 0

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{\frac{1}{2}}, \frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 39.1

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, \frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

   15. Applied rewrites 39.1%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, \frac{d \cdot \sqrt{\frac{c0}{h \cdot w}}}{D}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 2: 43.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)\\ t_1 := \frac{\left(c0 \cdot d\_m\right) \cdot d\_m}{t\_0}\\ t_2 := \frac{c0}{2 \cdot w}\\ t_3 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{t\_0}\\ \mathbf{if}\;t\_2 \cdot \left(t\_3 + \sqrt{t\_3 \cdot t\_3 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M\_m \cdot M\_m\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (* (* w h) (* D_m D_m)))
        (t_1 (/ (* (* c0 d_m) d_m) t_0))
        (t_2 (/ c0 (* 2.0 w)))
        (t_3 (/ (* c0 (* d_m d_m)) t_0)))
   (if (<= (* t_2 (+ t_3 (sqrt (- (* t_3 t_3) (* M_m M_m))))) INFINITY)
     (* t_2 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = (w * h) * (D_m * D_m);
	double t_1 = ((c0 * d_m) * d_m) / t_0;
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d_m * d_m)) / t_0;
	double tmp;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	} else {
		tmp = ((pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = (w * h) * (D_m * D_m);
	double t_1 = ((c0 * d_m) * d_m) / t_0;
	double t_2 = c0 / (2.0 * w);
	double t_3 = (c0 * (d_m * d_m)) / t_0;
	double tmp;
	if ((t_2 * (t_3 + Math.sqrt(((t_3 * t_3) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_2 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))));
	} else {
		tmp = ((Math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = (w * h) * (D_m * D_m)
	t_1 = ((c0 * d_m) * d_m) / t_0
	t_2 = c0 / (2.0 * w)
	t_3 = (c0 * (d_m * d_m)) / t_0
	tmp = 0
	if (t_2 * (t_3 + math.sqrt(((t_3 * t_3) - (M_m * M_m))))) <= math.inf:
		tmp = t_2 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))
	else:
		tmp = ((math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(Float64(w * h) * Float64(D_m * D_m))
	t_1 = Float64(Float64(Float64(c0 * d_m) * d_m) / t_0)
	t_2 = Float64(c0 / Float64(2.0 * w))
	t_3 = Float64(Float64(c0 * Float64(d_m * d_m)) / t_0)
	tmp = 0.0
	if (Float64(t_2 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_2 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m)))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	t_0 = (w * h) * (D_m * D_m);
	t_1 = ((c0 * d_m) * d_m) / t_0;
	t_2 = c0 / (2.0 * w);
	t_3 = (c0 * (d_m * d_m)) / t_0;
	tmp = 0.0;
	if ((t_2 * (t_3 + sqrt(((t_3 * t_3) - (M_m * M_m))))) <= Inf)
		tmp = t_2 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))));
	else
		tmp = (((-(M_m * M_m) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c0 * d$95$m), $MachinePrecision] * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$2 * 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], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 24.7

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

   3. Applied rewrites 24.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 24.9

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

   5. Applied rewrites 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 27.2

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

   7. Applied rewrites 27.2%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 3: 43.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{\left(D\_m \cdot D\_m\right) \cdot \left(h \cdot w\right)}\\ t_1 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \left(\left(d\_m \cdot d\_m\right) \cdot \left(t\_0 + {t\_0}^{1}\right)\right)}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M\_m \cdot M\_m\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* (* D_m D_m) (* h w))))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m)))))
        INFINITY)
     (* 0.5 (/ (* c0 (* (* d_m d_m) (+ t_0 (pow t_0 1.0)))) w))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / ((D_m * D_m) * (h * w));
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = 0.5 * ((c0 * ((d_m * d_m) * (t_0 + pow(t_0, 1.0)))) / w);
	} else {
		tmp = ((pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / ((D_m * D_m) * (h * w));
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if (((c0 / (2.0 * w)) * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * ((c0 * ((d_m * d_m) * (t_0 + Math.pow(t_0, 1.0)))) / w);
	} else {
		tmp = ((Math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = c0 / ((D_m * D_m) * (h * w))
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = 0.5 * ((c0 * ((d_m * d_m) * (t_0 + math.pow(t_0, 1.0)))) / w)
	else:
		tmp = ((math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 / Float64(Float64(D_m * D_m) * Float64(h * w)))
	t_1 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	tmp = 0.0
	if (Float64(Float64(c0 / Float64(2.0 * w)) * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(0.5 * Float64(Float64(c0 * Float64(Float64(d_m * d_m) * Float64(t_0 + (t_0 ^ 1.0)))) / w));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	t_0 = c0 / ((D_m * D_m) * (h * w));
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	tmp = 0.0;
	if (((c0 / (2.0 * w)) * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = 0.5 * ((c0 * ((d_m * d_m) * (t_0 + (t_0 ^ 1.0)))) / w);
	else
		tmp = (((-(M_m * M_m) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(0.5 * N[(N[(c0 * N[(N[(d$95$m * d$95$m), $MachinePrecision] * N[(t$95$0 + N[Power[t$95$0, 1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 33.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)} + {\left(\frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\right)}^{1}\right)\right)}{w}} \]

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 4: 41.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := c0 \cdot \left(d\_m \cdot d\_m\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ t_3 := t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right)\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot \mathsf{fma}\left(\frac{c0}{h}, \frac{d\_m \cdot d\_m}{w}, {\left(\frac{t\_0}{h \cdot w}\right)}^{1}\right)}{\left(D\_m \cdot D\_m\right) \cdot w}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot {\left(c0 \cdot \sqrt{\frac{d\_m \cdot d\_m}{\left(D\_m \cdot D\_m\right) \cdot \left(c0 \cdot \left(h \cdot w\right)\right)}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M\_m \cdot M\_m\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (* c0 (* d_m d_m)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* w h) (* D_m D_m))))
        (t_3 (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m)))))))
   (if (<= t_3 5e+64)
     (*
      0.5
      (/
       (* c0 (fma (/ c0 h) (/ (* d_m d_m) w) (pow (/ t_0 (* h w)) 1.0)))
       (* (* D_m D_m) w)))
     (if (<= t_3 INFINITY)
       (*
        t_1
        (pow (* c0 (sqrt (/ (* d_m d_m) (* (* D_m D_m) (* c0 (* h w)))))) 2.0))
       (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5)))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 * (d_m * d_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D_m * D_m));
	double t_3 = t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))));
	double tmp;
	if (t_3 <= 5e+64) {
		tmp = 0.5 * ((c0 * fma((c0 / h), ((d_m * d_m) / w), pow((t_0 / (h * w)), 1.0))) / ((D_m * D_m) * w));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * pow((c0 * sqrt(((d_m * d_m) / ((D_m * D_m) * (c0 * (h * w)))))), 2.0);
	} else {
		tmp = ((pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 * Float64(d_m * d_m))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D_m * D_m)))
	t_3 = Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m)))))
	tmp = 0.0
	if (t_3 <= 5e+64)
		tmp = Float64(0.5 * Float64(Float64(c0 * fma(Float64(c0 / h), Float64(Float64(d_m * d_m) / w), (Float64(t_0 / Float64(h * w)) ^ 1.0))) / Float64(Float64(D_m * D_m) * w)));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * (Float64(c0 * sqrt(Float64(Float64(d_m * d_m) / Float64(Float64(D_m * D_m) * Float64(c0 * Float64(h * w)))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 5e+64], N[(0.5 * N[(N[(c0 * N[(N[(c0 / h), $MachinePrecision] * N[(N[(d$95$m * d$95$m), $MachinePrecision] / w), $MachinePrecision] + N[Power[N[(t$95$0 / N[(h * w), $MachinePrecision]), $MachinePrecision], 1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(D$95$m * D$95$m), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[Power[N[(c0 * N[Sqrt[N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(c0 * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))))) < 5e64

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in D around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 31.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot \mathsf{fma}\left(\frac{c0}{h}, \frac{d \cdot d}{w}, {\left(\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}\right)}^{1}\right)}{\left(D \cdot D\right) \cdot w}} \]

   if 5e64 < (*.f64 (/.f64 c0 (*.f64 #s(literal 2 binary64) w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in c0 around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

   6. Applied rewrites 25.3%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 5: 40.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := c0 \cdot \left(d\_m \cdot d\_m\right)\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{t\_0}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\sqrt{M\_m}, \frac{\sqrt{\frac{t\_0}{h \cdot w}}}{D\_m}, c0 \cdot \left(\frac{d\_m}{\left(h \cdot w\right) \cdot D\_m} \cdot \frac{d\_m}{D\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M\_m \cdot M\_m\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (* c0 (* d_m d_m)))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ t_0 (* (* w h) (* D_m D_m)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M_m M_m))))) INFINITY)
     (*
      t_1
      (fma
       (sqrt M_m)
       (/ (sqrt (/ t_0 (* h w))) D_m)
       (* c0 (* (/ d_m (* (* h w) D_m)) (/ d_m D_m)))))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 * (d_m * d_m);
	double t_1 = c0 / (2.0 * w);
	double t_2 = t_0 / ((w * h) * (D_m * D_m));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_1 * fma(sqrt(M_m), (sqrt((t_0 / (h * w))) / D_m), (c0 * ((d_m / ((h * w) * D_m)) * (d_m / D_m))));
	} else {
		tmp = ((pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 * Float64(d_m * d_m))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(t_0 / Float64(Float64(w * h) * Float64(D_m * D_m)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M_m * M_m))))) <= Inf)
		tmp = Float64(t_1 * fma(sqrt(M_m), Float64(sqrt(Float64(t_0 / Float64(h * w))) / D_m), Float64(c0 * Float64(Float64(d_m / Float64(Float64(h * w) * D_m)) * Float64(d_m / D_m)))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 * N[(t$95$2 + N[Sqrt[N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[Sqrt[M$95$m], $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 / N[(h * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / D$95$m), $MachinePrecision] + N[(c0 * N[(N[(d$95$m / N[(N[(h * w), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(d$95$m / D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 32.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]

   5. Applied rewrites 32.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \color{blue}{\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 33.0

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}\right) - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 38.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{0.5}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \color{blue}{\frac{d}{D}}\right)\right) \]

   9. Applied rewrites 38.4%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right) - M\right)}^{0.5}, c0 \cdot \color{blue}{\left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)}\right) \]

   10. Taylor expanded in D around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 34.4

          \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, \frac{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}}}{D}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

   12. Applied rewrites 34.4%

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}, M\right)\right)}^{0.5}, \color{blue}{\frac{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}}}{D}}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]

   13. Taylor expanded in c0 around 0

       \[\leadsto \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\color{blue}{\sqrt{M}}, \frac{\sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{h \cdot w}}}{D}, c0 \cdot \left(\frac{d}{\left(h \cdot w\right) \cdot D} \cdot \frac{d}{D}\right)\right) \]
   14. Step-by-step derivation

       1. lower-sqrt.f64 33.8

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

   15. Applied rewrites 33.8%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 6: 39.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d\_m \cdot d\_m\right)}{\left(w \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M\_m \cdot M\_m}\right) \leq \infty:\\ \;\;\;\;t\_0 \cdot {\left(c0 \cdot \sqrt{\frac{d\_m \cdot d\_m}{\left(D\_m \cdot D\_m\right) \cdot \left(c0 \cdot \left(h \cdot w\right)\right)}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-M\_m \cdot M\_m\right)}^{0.5} \cdot c0}{w} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<= (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M_m M_m))))) INFINITY)
     (*
      t_0
      (pow (* c0 (sqrt (/ (* d_m d_m) (* (* D_m D_m) (* c0 (* h w)))))) 2.0))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= ((double) INFINITY)) {
		tmp = t_0 * pow((c0 * sqrt(((d_m * d_m) / ((D_m * D_m) * (c0 * (h * w)))))), 2.0);
	} else {
		tmp = ((pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	double tmp;
	if ((t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * Math.pow((c0 * Math.sqrt(((d_m * d_m) / ((D_m * D_m) * (c0 * (h * w)))))), 2.0);
	} else {
		tmp = ((Math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m))
	tmp = 0
	if (t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M_m * M_m))))) <= math.inf:
		tmp = t_0 * math.pow((c0 * math.sqrt(((d_m * d_m) / ((D_m * D_m) * (c0 * (h * w)))))), 2.0)
	else:
		tmp = ((math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	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(c0 * sqrt(Float64(Float64(d_m * d_m) / Float64(Float64(D_m * D_m) * Float64(c0 * Float64(h * w)))))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m));
	tmp = 0.0;
	if ((t_0 * (t_1 + sqrt(((t_1 * t_1) - (M_m * M_m))))) <= Inf)
		tmp = t_0 * ((c0 * sqrt(((d_m * d_m) / ((D_m * D_m) * (c0 * (h * w)))))) ^ 2.0);
	else
		tmp = (((-(M_m * M_m) ^ 0.5) * c0) / w) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $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[Power[N[(c0 * N[Sqrt[N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(c0 * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in c0 around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

   6. Applied rewrites 25.3%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 7: 38.0% accurate, 0.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (* 0.5 (/ (pow (* c0 d_m) 2.0) (* (* D_m D_m) (* h (* w w)))))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

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

Java

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

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= math.inf:
		tmp = 0.5 * (math.pow((c0 * d_m), 2.0) / ((D_m * D_m) * (h * (w * w))))
	else:
		tmp = ((math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	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(0.5 * Float64((Float64(c0 * d_m) ^ 2.0) / Float64(Float64(D_m * D_m) * Float64(h * Float64(w * w)))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $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[(0.5 * N[(N[Power[N[(c0 * d$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]

   7. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 24.8

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

   9. Applied rewrites 24.8%

      \[\leadsto 0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. pow2 N/A

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

       6. pow-prod-down N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-*.f64 30.7

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

   11. Applied rewrites 30.7%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 8: 37.6% accurate, 0.7× speedup

Math

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

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (let* ((t_0 (/ (* c0 (* d_m d_m)) (* (* w h) (* D_m D_m)))))
   (if (<=
        (* (/ c0 (* 2.0 w)) (+ t_0 (sqrt (- (* t_0 t_0) (* M_m M_m)))))
        INFINITY)
     (* 0.5 (/ (* (* c0 c0) (* d_m d_m)) (* (* D_m D_m) (* h (* w w)))))
     (* (/ (* (pow (- (* M_m M_m)) 0.5) c0) w) 0.5))))

C

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

Java

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

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	t_0 = (c0 * (d_m * d_m)) / ((w * h) * (D_m * D_m))
	tmp = 0
	if ((c0 / (2.0 * w)) * (t_0 + math.sqrt(((t_0 * t_0) - (M_m * M_m))))) <= math.inf:
		tmp = 0.5 * (((c0 * c0) * (d_m * d_m)) / ((D_m * D_m) * (h * (w * w))))
	else:
		tmp = ((math.pow(-(M_m * M_m), 0.5) * c0) / w) * 0.5
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	t_0 = Float64(Float64(c0 * Float64(d_m * d_m)) / Float64(Float64(w * h) * Float64(D_m * D_m)))
	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(0.5 * Float64(Float64(Float64(c0 * c0) * Float64(d_m * d_m)) / Float64(Float64(D_m * D_m) * Float64(h * Float64(w * w)))));
	else
		tmp = Float64(Float64(Float64((Float64(-Float64(M_m * M_m)) ^ 0.5) * c0) / w) * 0.5);
	end
	return tmp
end

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := Block[{t$95$0 = N[(N[(c0 * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D$95$m * D$95$m), $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[(0.5 * N[(N[(N[(c0 * c0), $MachinePrecision] * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[(-N[(M$95$m * M$95$m), $MachinePrecision]), 0.5], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]

   7. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 24.8

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

   9. Applied rewrites 24.8%

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

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

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      3. lift-neg.f64 N/A

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

      4. pow1/2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      5. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left({M}^{2}\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\mathsf{neg}\left(M \cdot M\right)\right)}^{\frac{1}{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.0

         \[\leadsto \frac{{\left(-M \cdot M\right)}^{0.5} \cdot c0}{w} \cdot 0.5 \]

   6. Applied rewrites 22.0%

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

Alternative 9: 32.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.06 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{-M\_m \cdot M\_m} \cdot c0}{w} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{c0 \cdot c0}{D\_m \cdot D\_m} \cdot \frac{d\_m \cdot d\_m}{h \cdot \left(w \cdot w\right)}\right)\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (if (<= M_m 1.06e-162)
   (* (/ (* (sqrt (- (* M_m M_m))) c0) w) 0.5)
   (* 0.5 (* (/ (* c0 c0) (* D_m D_m)) (/ (* d_m d_m) (* h (* w w)))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double tmp;
	if (M_m <= 1.06e-162) {
		tmp = ((sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
	} else {
		tmp = 0.5 * (((c0 * c0) / (D_m * D_m)) * ((d_m * d_m) / (h * (w * w))));
	}
	return tmp;
}

Fortran

D_m =     private
d_m =     private
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_m, d_m_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_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    real(8) :: tmp
    if (m_m <= 1.06d-162) then
        tmp = ((sqrt(-(m_m * m_m)) * c0) / w) * 0.5d0
    else
        tmp = 0.5d0 * (((c0 * c0) / (d_m * d_m)) * ((d_m_1 * d_m_1) / (h * (w * w))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double tmp;
	if (M_m <= 1.06e-162) {
		tmp = ((Math.sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
	} else {
		tmp = 0.5 * (((c0 * c0) / (D_m * D_m)) * ((d_m * d_m) / (h * (w * w))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	tmp = 0
	if M_m <= 1.06e-162:
		tmp = ((math.sqrt(-(M_m * M_m)) * c0) / w) * 0.5
	else:
		tmp = 0.5 * (((c0 * c0) / (D_m * D_m)) * ((d_m * d_m) / (h * (w * w))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	tmp = 0.0
	if (M_m <= 1.06e-162)
		tmp = Float64(Float64(Float64(sqrt(Float64(-Float64(M_m * M_m))) * c0) / w) * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 * c0) / Float64(D_m * D_m)) * Float64(Float64(d_m * d_m) / Float64(h * Float64(w * w)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	tmp = 0.0;
	if (M_m <= 1.06e-162)
		tmp = ((sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
	else
		tmp = 0.5 * (((c0 * c0) / (D_m * D_m)) * ((d_m * d_m) / (h * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := If[LessEqual[M$95$m, 1.06e-162], N[(N[(N[(N[Sqrt[(-N[(M$95$m * M$95$m), $MachinePrecision])], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[(N[(c0 * c0), $MachinePrecision] / N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.06000000000000003e-162

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]

   if 1.06000000000000003e-162 < M

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]

   7. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 24.8

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

   9. Applied rewrites 24.8%

      \[\leadsto 0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d \cdot d\right)}{\color{blue}{\left(D \cdot D\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. times-frac N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 N/A

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

       16. pow2 N/A

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

       17. lift-*.f64 N/A

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

       18. pow2 N/A

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

       19. lift-*.f64 N/A

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

       20. pow2 N/A

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

       21. lift-*.f64 N/A

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

       22. lift-*.f64 N/A

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

   11. Applied rewrites 24.8%

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

Alternative 10: 32.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.06 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{-M\_m \cdot M\_m} \cdot c0}{w} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\left(c0 \cdot c0\right) \cdot \left(d\_m \cdot d\_m\right)}{\left(D\_m \cdot D\_m\right) \cdot \left(h \cdot \left(w \cdot w\right)\right)}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (if (<= M_m 1.06e-162)
   (* (/ (* (sqrt (- (* M_m M_m))) c0) w) 0.5)
   (* 0.5 (/ (* (* c0 c0) (* d_m d_m)) (* (* D_m D_m) (* h (* w w)))))))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double tmp;
	if (M_m <= 1.06e-162) {
		tmp = ((sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
	} else {
		tmp = 0.5 * (((c0 * c0) * (d_m * d_m)) / ((D_m * D_m) * (h * (w * w))));
	}
	return tmp;
}

Fortran

D_m =     private
d_m =     private
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_m, d_m_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_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    real(8) :: tmp
    if (m_m <= 1.06d-162) then
        tmp = ((sqrt(-(m_m * m_m)) * c0) / w) * 0.5d0
    else
        tmp = 0.5d0 * (((c0 * c0) * (d_m_1 * d_m_1)) / ((d_m * d_m) * (h * (w * w))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	double tmp;
	if (M_m <= 1.06e-162) {
		tmp = ((Math.sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
	} else {
		tmp = 0.5 * (((c0 * c0) * (d_m * d_m)) / ((D_m * D_m) * (h * (w * w))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	tmp = 0
	if M_m <= 1.06e-162:
		tmp = ((math.sqrt(-(M_m * M_m)) * c0) / w) * 0.5
	else:
		tmp = 0.5 * (((c0 * c0) * (d_m * d_m)) / ((D_m * D_m) * (h * (w * w))))
	return tmp

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	tmp = 0.0
	if (M_m <= 1.06e-162)
		tmp = Float64(Float64(Float64(sqrt(Float64(-Float64(M_m * M_m))) * c0) / w) * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(c0 * c0) * Float64(d_m * d_m)) / Float64(Float64(D_m * D_m) * Float64(h * Float64(w * w)))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp_2 = code(c0, w, h, D_m, d_m, M_m)
	tmp = 0.0;
	if (M_m <= 1.06e-162)
		tmp = ((sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
	else
		tmp = 0.5 * (((c0 * c0) * (d_m * d_m)) / ((D_m * D_m) * (h * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := If[LessEqual[M$95$m, 1.06e-162], N[(N[(N[(N[Sqrt[(-N[(M$95$m * M$95$m), $MachinePrecision])], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[(N[(c0 * c0), $MachinePrecision] * N[(d$95$m * d$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.06000000000000003e-162

   1. Initial program 25.1%

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

   2. Taylor expanded in c0 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

      8. pow2 N/A

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

      9. lift-*.f64 14.5

         \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

   4. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]

   if 1.06000000000000003e-162 < M

   1. Initial program 25.1%

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

   3. Applied rewrites 32.5%

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c0, \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}, M\right)\right)}^{0.5}, {\left(c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D} - M\right)}^{0.5}, c0 \cdot \frac{d \cdot d}{\left(\left(h \cdot w\right) \cdot D\right) \cdot D}\right)} \]

   4. Taylor expanded in h around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c0 \cdot {\left(\frac{c0 \cdot \left(\left(d \cdot d\right) \cdot h\right)}{\left(D \cdot D\right) \cdot w}\right)}^{1}}{\left(h \cdot h\right) \cdot w}} \]

   7. Taylor expanded in c0 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 24.8

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

   9. Applied rewrites 24.8%

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

Alternative 11: 14.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \frac{\sqrt{-M\_m \cdot M\_m} \cdot c0}{w} \cdot 0.5 \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (* (/ (* (sqrt (- (* M_m M_m))) c0) w) 0.5))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	return ((sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
}

Fortran

D_m =     private
d_m =     private
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_m, d_m_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_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    code = ((sqrt(-(m_m * m_m)) * c0) / w) * 0.5d0
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	return ((Math.sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	return ((math.sqrt(-(M_m * M_m)) * c0) / w) * 0.5

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	return Float64(Float64(Float64(sqrt(Float64(-Float64(M_m * M_m))) * c0) / w) * 0.5)
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp = code(c0, w, h, D_m, d_m, M_m)
	tmp = ((sqrt(-(M_m * M_m)) * c0) / w) * 0.5;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := N[(N[(N[(N[Sqrt[(-N[(M$95$m * M$95$m), $MachinePrecision])], $MachinePrecision] * c0), $MachinePrecision] / w), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in c0 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-neg.f64 N/A

      \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot c0}{w} \cdot \frac{1}{2} \]

   8. pow2 N/A

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot \frac{1}{2} \]

   9. lift-*.f64 14.5

      \[\leadsto \frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5 \]

4. Applied rewrites 14.5%

   \[\leadsto \color{blue}{\frac{\sqrt{-M \cdot M} \cdot c0}{w} \cdot 0.5} \]
5. Add Preprocessing

Alternative 12: 0.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ d_m = \left|d\right| \\ M_m = \left|M\right| \\ \frac{c0}{2 \cdot w} \cdot \left(\sqrt{-1} \cdot M\_m\right) \end{array} \]

FPCore

D_m = (fabs.f64 D)
d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
(FPCore (c0 w h D_m d_m M_m)
 :precision binary64
 (* (/ c0 (* 2.0 w)) (* (sqrt -1.0) M_m)))

C

D_m = fabs(D);
d_m = fabs(d);
M_m = fabs(M);
double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	return (c0 / (2.0 * w)) * (sqrt(-1.0) * M_m);
}

Fortran

D_m =     private
d_m =     private
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_m, d_m_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_m
    real(8), intent (in) :: d_m_1
    real(8), intent (in) :: m_m
    code = (c0 / (2.0d0 * w)) * (sqrt((-1.0d0)) * m_m)
end function

Java

D_m = Math.abs(D);
d_m = Math.abs(d);
M_m = Math.abs(M);
public static double code(double c0, double w, double h, double D_m, double d_m, double M_m) {
	return (c0 / (2.0 * w)) * (Math.sqrt(-1.0) * M_m);
}

Python

D_m = math.fabs(D)
d_m = math.fabs(d)
M_m = math.fabs(M)
def code(c0, w, h, D_m, d_m, M_m):
	return (c0 / (2.0 * w)) * (math.sqrt(-1.0) * M_m)

Julia

D_m = abs(D)
d_m = abs(d)
M_m = abs(M)
function code(c0, w, h, D_m, d_m, M_m)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(sqrt(-1.0) * M_m))
end

MATLAB

D_m = abs(D);
d_m = abs(d);
M_m = abs(M);
function tmp = code(c0, w, h, D_m, d_m, M_m)
	tmp = (c0 / (2.0 * w)) * (sqrt(-1.0) * M_m);
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
code[c0_, w_, h_, D$95$m_, d$95$m_, M$95$m_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[-1.0], $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in M around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 0.0

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

4. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025127 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))