Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.5% → 43.6%

Time: 9.6s

Alternatives: 17

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 43.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{\left(D \cdot D\right) \cdot \left(h \cdot w\right)}\\ t_1 := \frac{c0}{2 \cdot w}\\ t_2 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ \mathbf{if}\;t\_1 \cdot \left(t\_2 + \sqrt{t\_2 \cdot t\_2 - M \cdot M}\right) \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\left(d \cdot d\right) \cdot \left(t\_0 + {t\_0}^{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* (* D D) (* h w))))
        (t_1 (/ c0 (* 2.0 w)))
        (t_2 (/ (* c0 (* d d)) (* (* w h) (* D D)))))
   (if (<= (* t_1 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_1 (* (* d d) (+ t_0 (pow t_0 1.0))))
     (* 0.5 (/ (* c0 (pow (- (* M M)) 0.5)) w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / ((D * D) * (h * w));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_1 * ((d * d) * (t_0 + pow(t_0, 1.0)));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / ((D * D) * (h * w));
	double t_1 = c0 / (2.0 * w);
	double t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	double tmp;
	if ((t_1 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 * ((d * d) * (t_0 + Math.pow(t_0, 1.0)));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / ((D * D) * (h * w))
	t_1 = c0 / (2.0 * w)
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D))
	tmp = 0
	if (t_1 * (t_2 + math.sqrt(((t_2 * t_2) - (M * M))))) <= math.inf:
		tmp = t_1 * ((d * d) * (t_0 + math.pow(t_0, 1.0)))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(Float64(D * D) * Float64(h * w)))
	t_1 = Float64(c0 / Float64(2.0 * w))
	t_2 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	tmp = 0.0
	if (Float64(t_1 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_1 * Float64(Float64(d * d) * Float64(t_0 + (t_0 ^ 1.0))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / ((D * D) * (h * w));
	t_1 = c0 / (2.0 * w);
	t_2 = (c0 * (d * d)) / ((w * h) * (D * D));
	tmp = 0.0;
	if ((t_1 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_1 * ((d * d) * (t_0 + (t_0 ^ 1.0)));
	else
		tmp = 0.5 * ((c0 * (-(M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $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 * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * N[(N[(d * d), $MachinePrecision] * N[(t$95$0 + N[Power[t$95$0, 1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-pow2 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

   8. Applied rewrites 32.4%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 2: 43.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left({d}^{2} \cdot \left(\frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)} + {\left(\sqrt{\frac{c0}{{D}^{2} \cdot \left(h \cdot w\right)}}\right)}^{2}\right)\right)}{w}} \]
   7. 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

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

   8. Applied rewrites 32.3%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 3: 41.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in D around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c0 \cdot \left(\frac{c0 \cdot {d}^{2}}{h \cdot w} + {\left(\sqrt{\frac{c0 \cdot {d}^{2}}{h \cdot w}}\right)}^{2}\right)}{{D}^{2} \cdot w}} \]
   7. 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

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

   8. Applied rewrites 30.4%

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

   if +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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 4: 39.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(M * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.1%

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

   4. Taylor expanded in c0 around 0

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

   6. Applied rewrites 14.6%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 5: 39.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{-1 \cdot \frac{c0}{\left(D \cdot D\right) \cdot \sqrt{h \cdot h}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \frac{\frac{\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}}{D \cdot D}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -2e-30)
     (*
      t_0
      (*
       (* d d)
       (+
        (/ (* -1.0 (/ c0 (* (* D D) (sqrt (* h h))))) w)
        (/ c0 (* (* (* D D) h) w)))))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (/ (/ (+ (sqrt (/ (* c0 c0) (* w w))) (/ c0 w)) (* D D)) h)))
       (* 0.5 (/ (* c0 (pow (- (* M M)) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-30) {
		tmp = t_0 * ((d * d) * (((-1.0 * (c0 / ((D * D) * sqrt((h * h))))) / w) + (c0 / (((D * D) * h) * w))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * (((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-30) {
		tmp = t_0 * ((d * d) * (((-1.0 * (c0 / ((D * D) * Math.sqrt((h * h))))) / w) + (c0 / (((D * D) * h) * w))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * (((Math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -2e-30:
		tmp = t_0 * ((d * d) * (((-1.0 * (c0 / ((D * D) * math.sqrt((h * h))))) / w) + (c0 / (((D * D) * h) * w))))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * (((math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -2e-30)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(Float64(-1.0 * Float64(c0 / Float64(Float64(D * D) * sqrt(Float64(h * h))))) / w) + Float64(c0 / Float64(Float64(Float64(D * D) * h) * w)))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(Float64(sqrt(Float64(Float64(c0 * c0) / Float64(w * w))) + Float64(c0 / w)) / Float64(D * D)) / h)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -2e-30)
		tmp = t_0 * ((d * d) * (((-1.0 * (c0 / ((D * D) * sqrt((h * h))))) / w) + (c0 / (((D * D) * h) * w))));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * (((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	else
		tmp = 0.5 * ((c0 * (-(M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-30], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[(-1.0 * N[(c0 / N[(N[(D * D), $MachinePrecision] * N[Sqrt[N[(h * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision] + N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $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))))) < -2e-30

   1. Initial program 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      8. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot \left(h \cdot h\right)}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      9. lift-*.f64 11.9

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot \left(h \cdot h\right)}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

   7. Applied rewrites 11.9%

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

   8. Taylor expanded in c0 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-sqrt.f64 14.6

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

   10. Applied rewrites 14.6%

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

   11. Taylor expanded in D around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 16.6

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

   13. Applied rewrites 16.6%

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

   if -2e-30 < (*.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 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {w}^{2}}} + \frac{c0}{{D}^{2} \cdot w}}{h}\right) \]

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 18.0

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

   10. Applied rewrites 18.0%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 6: 35.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision] * N[(N[(d * d), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(M * N[(N[(c0 * N[(N[(d * d), $MachinePrecision] / N[(N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.1%

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

   4. Taylor expanded in c0 around 0

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

   6. Applied rewrites 14.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 14.5%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 7: 35.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;t\_0 \cdot {\left(c0 \cdot \sqrt{\frac{d \cdot d}{\left(D \cdot D\right) \cdot \left(c0 \cdot \left(h \cdot w\right)\right)}}\right)}^{2}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \frac{\frac{\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}}{D \cdot D}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -2e-30)
     (* t_0 (pow (* c0 (sqrt (/ (* d d) (* (* D D) (* c0 (* h w)))))) 2.0))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (/ (/ (+ (sqrt (/ (* c0 c0) (* w w))) (/ c0 w)) (* D D)) h)))
       (* 0.5 (/ (* c0 (pow (- (* M M)) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-30) {
		tmp = t_0 * pow((c0 * sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))), 2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * (((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-30) {
		tmp = t_0 * Math.pow((c0 * Math.sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))), 2.0);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * (((Math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -2e-30:
		tmp = t_0 * math.pow((c0 * math.sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))), 2.0)
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * (((math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -2e-30)
		tmp = Float64(t_0 * (Float64(c0 * sqrt(Float64(Float64(d * d) / Float64(Float64(D * D) * Float64(c0 * Float64(h * w)))))) ^ 2.0));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(Float64(sqrt(Float64(Float64(c0 * c0) / Float64(w * w))) + Float64(c0 / w)) / Float64(D * D)) / h)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -2e-30)
		tmp = t_0 * ((c0 * sqrt(((d * d) / ((D * D) * (c0 * (h * w)))))) ^ 2.0);
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * (((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	else
		tmp = 0.5 * ((c0 * (-(M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-30], N[(t$95$0 * N[Power[N[(c0 * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(c0 * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $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))))) < -2e-30

   1. Initial program 24.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. difference-of-squares N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 30.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in c0 around inf

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left({c0}^{2} \cdot {\left(\sqrt{\frac{{d}^{2}}{{D}^{2} \cdot \left(c0 \cdot \left(h \cdot w\right)\right)}}\right)}^{2}\right)} \]
   7. 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

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

   8. Applied rewrites 24.9%

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

   if -2e-30 < (*.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 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {w}^{2}}} + \frac{c0}{{D}^{2} \cdot w}}{h}\right) \]

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 18.0

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

   10. Applied rewrites 18.0%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 8: 35.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(N[(h * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 21.4%

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

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

   1. Initial program 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 9: 33.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(N[(N[(d * d), $MachinePrecision] * N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(N[(h * h), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 21.5%

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

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

   1. Initial program 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 10: 32.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[Sqrt[N[(c0 * c0), $MachinePrecision]], $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * N[(h * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c0 / N[(N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $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))))) < -inf.0

   1. Initial program 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in w around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      4. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot {h}^{2}}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      8. pow2 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot \left(h \cdot h\right)}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

      9. lift-*.f64 11.9

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \left(\frac{\sqrt{\frac{c0 \cdot c0}{{D}^{4} \cdot \left(h \cdot h\right)}}}{w} + \frac{c0}{\left(\left(D \cdot D\right) \cdot h\right) \cdot w}\right)\right) \]

   7. Applied rewrites 11.9%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 12.9

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

   10. Applied rewrites 12.9%

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

   11. Taylor expanded in h around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 16.1

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

   13. Applied rewrites 16.1%

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

   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))))) < +inf.0

   1. Initial program 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {w}^{2}}} + \frac{c0}{{D}^{2} \cdot w}}{h}\right) \]

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 18.0

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

   10. Applied rewrites 18.0%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 11: 30.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;t\_0 \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) + \sqrt{-1 \cdot \left(M \cdot M\right)}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \frac{\frac{\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}}{D \cdot D}}{h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 -2e-30)
     (* t_0 (+ (* c0 (* (/ d (* (* w h) D)) (/ d D))) (sqrt (* -1.0 (* M M)))))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (/ (/ (+ (sqrt (/ (* c0 c0) (* w w))) (/ c0 w)) (* D D)) h)))
       (* 0.5 (/ (* c0 (pow (- (* M M)) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-30) {
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + sqrt((-1.0 * (M * M))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * (((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= -2e-30) {
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + Math.sqrt((-1.0 * (M * M))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * (((Math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= -2e-30:
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + math.sqrt((-1.0 * (M * M))))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * (((math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= -2e-30)
		tmp = Float64(t_0 * Float64(Float64(c0 * Float64(Float64(d / Float64(Float64(w * h) * D)) * Float64(d / D))) + sqrt(Float64(-1.0 * Float64(M * M)))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(Float64(sqrt(Float64(Float64(c0 * c0) / Float64(w * w))) + Float64(c0 / w)) / Float64(D * D)) / h)));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= -2e-30)
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + sqrt((-1.0 * (M * M))));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * (((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / (D * D)) / h));
	else
		tmp = 0.5 * ((c0 * (-(M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-30], N[(t$95$0 * N[(N[(c0 * N[(N[(d / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(-1.0 * N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / w), $MachinePrecision]), $MachinePrecision] / N[(D * D), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $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))))) < -2e-30

   1. Initial program 24.5%

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

   3. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 27.3

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

   5. Applied rewrites 27.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 34.5

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 11.0

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

   10. Applied rewrites 11.0%

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

   if -2e-30 < (*.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 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {w}^{2}}} + \frac{c0}{{D}^{2} \cdot w}}{h}\right) \]

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 18.0

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

   10. Applied rewrites 18.0%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 12: 30.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t\_0 \cdot \left(t\_1 + \sqrt{t\_1 \cdot t\_1 - M \cdot M}\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;t\_0 \cdot \left(c0 \cdot \left(\frac{d}{\left(w \cdot h\right) \cdot D} \cdot \frac{d}{D}\right) + \sqrt{-1 \cdot \left(M \cdot M\right)}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \left(\left(d \cdot d\right) \cdot \frac{\sqrt{\frac{c0 \cdot c0}{w \cdot w}} + \frac{c0}{w}}{\left(D \cdot D\right) \cdot h}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c0 \cdot {\left(-M \cdot M\right)}^{0.5}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M)))))))
   (if (<= t_2 0.0)
     (* t_0 (+ (* c0 (* (/ d (* (* w h) D)) (/ d D))) (sqrt (* -1.0 (* M M)))))
     (if (<= t_2 INFINITY)
       (*
        t_0
        (*
         (* d d)
         (/ (+ (sqrt (/ (* c0 c0) (* w w))) (/ c0 w)) (* (* D D) h))))
       (* 0.5 (/ (* c0 (pow (- (* M M)) 0.5)) w))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + sqrt((-1.0 * (M * M))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * ((d * d) * ((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / ((D * D) * h)));
	} else {
		tmp = 0.5 * ((c0 * pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + Math.sqrt(((t_1 * t_1) - (M * M))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + Math.sqrt((-1.0 * (M * M))));
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * ((d * d) * ((Math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / ((D * D) * h)));
	} else {
		tmp = 0.5 * ((c0 * Math.pow(-(M * M), 0.5)) / w);
	}
	return tmp;
}

Python

def code(c0, w, h, D, d, M):
	t_0 = c0 / (2.0 * w)
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D))
	t_2 = t_0 * (t_1 + math.sqrt(((t_1 * t_1) - (M * M))))
	tmp = 0
	if t_2 <= 0.0:
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + math.sqrt((-1.0 * (M * M))))
	elif t_2 <= math.inf:
		tmp = t_0 * ((d * d) * ((math.sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / ((D * D) * h)))
	else:
		tmp = 0.5 * ((c0 * math.pow(-(M * M), 0.5)) / w)
	return tmp

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))
	t_2 = Float64(t_0 * Float64(t_1 + sqrt(Float64(Float64(t_1 * t_1) - Float64(M * M)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(t_0 * Float64(Float64(c0 * Float64(Float64(d / Float64(Float64(w * h) * D)) * Float64(d / D))) + sqrt(Float64(-1.0 * Float64(M * M)))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * Float64(Float64(d * d) * Float64(Float64(sqrt(Float64(Float64(c0 * c0) / Float64(w * w))) + Float64(c0 / w)) / Float64(Float64(D * D) * h))));
	else
		tmp = Float64(0.5 * Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c0, w, h, D, d, M)
	t_0 = c0 / (2.0 * w);
	t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	t_2 = t_0 * (t_1 + sqrt(((t_1 * t_1) - (M * M))));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = t_0 * ((c0 * ((d / ((w * h) * D)) * (d / D))) + sqrt((-1.0 * (M * M))));
	elseif (t_2 <= Inf)
		tmp = t_0 * ((d * d) * ((sqrt(((c0 * c0) / (w * w))) + (c0 / w)) / ((D * D) * h)));
	else
		tmp = 0.5 * ((c0 * (-(M * M) ^ 0.5)) / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(t$95$1 + N[Sqrt[N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(t$95$0 * N[(N[(c0 * N[(N[(d / N[(N[(w * h), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(-1.0 * N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[(N[(d * d), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(c0 * c0), $MachinePrecision] / N[(w * w), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(c0 / w), $MachinePrecision]), $MachinePrecision] / N[(N[(D * D), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $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))))) < -0.0

   1. Initial program 24.5%

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

   3. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 27.3

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

   5. Applied rewrites 27.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 34.5

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 11.0

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

   10. Applied rewrites 11.0%

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

   if -0.0 < (*.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))))) < +inf.0

   1. Initial program 24.5%

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

   2. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(\left(d \cdot d\right) \cdot \frac{\sqrt{\frac{{c0}^{2}}{{D}^{4} \cdot {w}^{2}}} + \frac{c0}{{D}^{2} \cdot w}}{h}\right) \]

   7. Applied rewrites 13.2%

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

   8. Taylor expanded in D around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 17.0

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

   10. Applied rewrites 17.0%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 13: 27.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

   3. Applied rewrites 28.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 27.3

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

   5. Applied rewrites 27.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 34.5

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 11.0

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

   10. Applied rewrites 11.0%

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

   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 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 14: 27.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 24.5%

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

   3. Applied rewrites 26.7%

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

   4. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 7.8

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

   6. Applied rewrites 7.8%

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

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

   1. Initial program 24.5%

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

   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. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 14.9

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

   4. Applied rewrites 14.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-*.f64 22.2

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

   6. Applied rewrites 22.2%

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

Alternative 15: 22.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * (-(m * m) ** 0.5d0)) / w)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.5%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 14.9

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

4. Applied rewrites 14.9%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-neg.f64 N/A

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

   9. lift-*.f64 22.2

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

6. Applied rewrites 22.2%

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

Alternative 16: 14.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c0 \cdot \sqrt{-M \cdot M}}{w} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = 0.5d0 * ((c0 * sqrt(-(m * m))) / w)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.5%

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

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. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 14.9

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

4. Applied rewrites 14.9%

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

Alternative 17: 0.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{c0}{2 \cdot w} \cdot \left(M \cdot \sqrt{-1}\right) \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (* (/ c0 (* 2.0 w)) (* M (sqrt -1.0))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (M * sqrt(-1.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c0, w, h, d, d_1, m)
use fmin_fmax_functions
    real(8), intent (in) :: c0
    real(8), intent (in) :: w
    real(8), intent (in) :: h
    real(8), intent (in) :: d
    real(8), intent (in) :: d_1
    real(8), intent (in) :: m
    code = (c0 / (2.0d0 * w)) * (m * sqrt((-1.0d0)))
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (M * Math.sqrt(-1.0));
}

Python

def code(c0, w, h, D, d, M):
	return (c0 / (2.0 * w)) * (M * math.sqrt(-1.0))

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(M * sqrt(-1.0)))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = (c0 / (2.0 * w)) * (M * sqrt(-1.0));
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(M * N[Sqrt[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.5%

   \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

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. lower-*.f64 N/A

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(M \cdot \color{blue}{\sqrt{-1}}\right) \]

   2. lower-sqrt.f64 0.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \left(M \cdot \sqrt{-1}\right) \]

4. Applied rewrites 0.0%

   \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(M \cdot \sqrt{-1}\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025136 
(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))))))