Henrywood and Agarwal, Equation (13)

Percentage Accurate: 24.9% → 45.0%

Time: 8.5s

Alternatives: 8

Speedup: 0.6×

• could not determine a ground truth

  1. c0 = 1.1168990699626555e+193
  2. w = -1.3802034385884935e-297
  3. h = 4.3504560688140145e-221
  4. D = -1.14571608483402e-304
  5. d = -3.366071046765619e+274
  6. M = 1.500361313805147e-102

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: 24.9% 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: 45.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ c0 (* 2.0 w)))
        (t_1 (* c0 (* d d)))
        (t_2 (/ t_1 (* (* w h) (* D D))))
        (t_3 (/ t_1 (* (* D (* h w)) D))))
   (if (<= (* t_0 (+ t_2 (sqrt (- (* t_2 t_2) (* M M))))) INFINITY)
     (* t_0 (+ t_3 (sqrt (- (* t_3 t_3) (* M M)))))
     (/ (* c0 (pow (- (* M M)) 0.5)) (+ w 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);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= ((double) INFINITY)) {
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	} else {
		tmp = (c0 * pow(-(M * M), 0.5)) / (w + 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);
	double t_2 = t_1 / ((w * h) * (D * D));
	double t_3 = t_1 / ((D * (h * w)) * D);
	double tmp;
	if ((t_0 * (t_2 + Math.sqrt(((t_2 * t_2) - (M * M))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 * (t_3 + Math.sqrt(((t_3 * t_3) - (M * M))));
	} else {
		tmp = (c0 * Math.pow(-(M * M), 0.5)) / (w + w);
	}
	return tmp;
}

Python

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

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(c0 / Float64(2.0 * w))
	t_1 = Float64(c0 * Float64(d * d))
	t_2 = Float64(t_1 / Float64(Float64(w * h) * Float64(D * D)))
	t_3 = Float64(t_1 / Float64(Float64(D * Float64(h * w)) * D))
	tmp = 0.0
	if (Float64(t_0 * Float64(t_2 + sqrt(Float64(Float64(t_2 * t_2) - Float64(M * M))))) <= Inf)
		tmp = Float64(t_0 * Float64(t_3 + sqrt(Float64(Float64(t_3 * t_3) - Float64(M * M)))));
	else
		tmp = Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / Float64(w + 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);
	t_2 = t_1 / ((w * h) * (D * D));
	t_3 = t_1 / ((D * (h * w)) * D);
	tmp = 0.0;
	if ((t_0 * (t_2 + sqrt(((t_2 * t_2) - (M * M))))) <= Inf)
		tmp = t_0 * (t_3 + sqrt(((t_3 * t_3) - (M * M))));
	else
		tmp = (c0 * (-(M * M) ^ 0.5)) / (w + 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[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(D * N[(h * w), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * 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$0 * N[(t$95$3 + N[Sqrt[N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / N[(w + 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 24.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.7

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

   3. Applied rewrites 24.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 27.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 27.7

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

   7. Applied rewrites 27.7%

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

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

   1. Initial program 24.9%

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

   2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-pow.f64 14.0

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

   9. Applied rewrites 14.0%

      \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-neg.f64 21.5

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

   11. Applied rewrites 21.5%

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

Alternative 2: 44.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(d * c0), $MachinePrecision] / N[(N[(N[(h * N[(D / d), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * D), $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[(N[(c0 / N[(w + w), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / N[(w + 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 24.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 24.4

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

   3. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 24.6

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

   5. Applied rewrites 24.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 35.2

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

   7. Applied rewrites 35.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. div-flip N/A

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

      6. frac-times N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 34.1

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

   9. Applied rewrites 34.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. div-flip N/A

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

       6. frac-times N/A

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

       7. *-commutative N/A

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

       8. *-rgt-identity N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 34.4

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

   11. Applied rewrites 34.4%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. div-flip N/A

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

       6. frac-times N/A

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

       7. *-commutative N/A

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

       8. *-rgt-identity N/A

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

       9. lower-/.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 37.3

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

   13. Applied rewrites 37.3%

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

       1. lift-*.f64 N/A

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

       2. count-2-rev N/A

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

       3. lift-+.f64 37.3

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

   15. Applied rewrites 37.3%

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

   17. Applied rewrites 36.1%

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

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

   1. Initial program 24.9%

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

   2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-pow.f64 14.0

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

   9. Applied rewrites 14.0%

      \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-neg.f64 21.5

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

   11. Applied rewrites 21.5%

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

Alternative 3: 44.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $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[(t$95$0 + N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / N[(w + 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 24.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 24.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 23.9

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

   3. Applied rewrites 25.9%

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

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

   1. Initial program 24.9%

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

   2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-pow.f64 14.0

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

   9. Applied rewrites 14.0%

      \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-neg.f64 21.5

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

   11. Applied rewrites 21.5%

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

Alternative 4: 43.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (* (/ (* d d) (* (* (* D D) w) h)) c0))
        (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 (fma 1.0 (fabs t_0) t_0))
     (/ (* c0 (pow (- (* M M)) 0.5)) (+ w w)))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = ((d * d) / (((D * D) * w) * h)) * c0;
	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 * fma(1.0, fabs(t_0), t_0);
	} else {
		tmp = (c0 * pow(-(M * M), 0.5)) / (w + w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(Float64(d * d) / Float64(Float64(Float64(D * D) * w) * h)) * c0)
	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 * fma(1.0, abs(t_0), t_0));
	else
		tmp = Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / Float64(w + w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(N[(d * d), $MachinePrecision] / N[(N[(N[(D * D), $MachinePrecision] * w), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * c0), $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[(1.0 * N[Abs[t$95$0], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / N[(w + 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 24.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   3. Applied rewrites 26.2%

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

   4. Taylor expanded in c0 around inf

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

   6. Applied rewrites 34.0%

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

   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 24.9%

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

   2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-pow.f64 14.0

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

   9. Applied rewrites 14.0%

      \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-neg.f64 21.5

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

   11. Applied rewrites 21.5%

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

Alternative 5: 26.6% accurate, 0.6× speedup

Math

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

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 (* -1.0 (pow M 2.0)))))
     (/ (* c0 (pow (- (* M M)) 0.5)) (+ w 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((-1.0 * pow(M, 2.0))));
	} else {
		tmp = (c0 * pow(-(M * M), 0.5)) / (w + 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((-1.0 * Math.pow(M, 2.0))));
	} else {
		tmp = (c0 * Math.pow(-(M * M), 0.5)) / (w + 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((-1.0 * math.pow(M, 2.0))))
	else:
		tmp = (c0 * math.pow(-(M * M), 0.5)) / (w + 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(-1.0 * (M ^ 2.0)))));
	else
		tmp = Float64(Float64(c0 * (Float64(-Float64(M * M)) ^ 0.5)) / Float64(w + 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((-1.0 * (M ^ 2.0))));
	else
		tmp = (c0 * (-(M * M) ^ 0.5)) / (w + 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[(-1.0 * N[Power[M, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 * N[Power[(-N[(M * M), $MachinePrecision]), 0.5], $MachinePrecision]), $MachinePrecision] / N[(w + 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 24.9%

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

   2. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 7.5

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

   4. Applied rewrites 7.5%

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

   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 24.9%

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

   2. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in c0 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-pow.f64 14.0

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

   9. Applied rewrites 14.0%

      \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
   10. Step-by-step derivation

       1. lift-sqrt.f64 N/A

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

       2. pow1/2 N/A

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

       3. lift-neg.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-neg.f64 21.5

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

   11. Applied rewrites 21.5%

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

Alternative 6: 21.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 * pow(-(M * M), 0.5)) / (w + 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 = (c0 * (-(m * m) ** 0.5d0)) / (w + w)
end function

Java

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

Python

def code(c0, w, h, D, d, M):
	return (c0 * math.pow(-(M * M), 0.5)) / (w + w)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 0.0%

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

7. Taylor expanded in c0 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-pow.f64 14.0

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

9. Applied rewrites 14.0%

   \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
10. Step-by-step derivation

    1. lift-sqrt.f64 N/A

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

    2. pow1/2 N/A

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

    3. lift-neg.f64 N/A

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

    4. lift-pow.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. lower-pow.f64 N/A

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

    8. lower-neg.f64 21.5

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

11. Applied rewrites 21.5%

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

Alternative 7: 14.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{-M \cdot M} \cdot c0}{w + w} \end{array} \]

FPCore

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

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (sqrt(-(M * M)) * c0) / (w + 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 = (sqrt(-(m * m)) * c0) / (w + w)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 0.0%

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

7. Taylor expanded in c0 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-pow.f64 14.0

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

9. Applied rewrites 14.0%

   \[\leadsto \frac{\color{blue}{c0 \cdot \sqrt{-{M}^{2}}}}{w + w} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{c0 \cdot \color{blue}{\sqrt{-{M}^{2}}}}{w + w} \]

    2. *-commutative N/A

       \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot \color{blue}{c0}}{w + w} \]

    3. lower-*.f64 14.0

       \[\leadsto \frac{\sqrt{-{M}^{2}} \cdot \color{blue}{c0}}{w + w} \]

    4. lift-pow.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 14.0

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

11. Applied rewrites 14.0%

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

Alternative 8: 0.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\sqrt{-1} \cdot M\right) \cdot c0}{w + w} \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (/ (* (* (sqrt -1.0) M) c0) (+ w w)))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return ((sqrt(-1.0) * M) * c0) / (w + 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 = ((sqrt((-1.0d0)) * m) * c0) / (w + w)
end function

Java

public static double code(double c0, double w, double h, double D, double d, double M) {
	return ((Math.sqrt(-1.0) * M) * c0) / (w + w);
}

Python

def code(c0, w, h, D, d, M):
	return ((math.sqrt(-1.0) * M) * c0) / (w + w)

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(Float64(sqrt(-1.0) * M) * c0) / Float64(w + w))
end

MATLAB

function tmp = code(c0, w, h, D, d, M)
	tmp = ((sqrt(-1.0) * M) * c0) / (w + w);
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(N[(N[Sqrt[-1.0], $MachinePrecision] * M), $MachinePrecision] * c0), $MachinePrecision] / N[(w + w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.9%

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

2. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2025151 
(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))))))